Axiomatic mathematics has no foundation
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Re: Axiomatic mathematics has no foundation
I was starting to write out another long response but you've lost me.
"We can't define axioms"
I'm not sure what you mean by "define" but what we certainly can do is state axioms in some chosen upon language. I don't know what your issue is with this. It's just some symbols on paper. Here they are.
They're right there, stated with all terms defined. What more do you want? It's written in logic languages and all of the details are out of my expertise but it seems like you may want to look into a textbook about formal languages and see if that addresses your concerns at all. There are all sorts of things trolling around Wikipedia that sound like the things you are talking about.
Sorry to sort of give up but you have all of this set up in your head differently than I have it set up and I think everyone is just talking past eachother at this point since we can't explain our ideas in the other person's paradigm. I want to encourage you to realize that you're probably not the first person to think about this and that there probably exists a thorough discussion of it somewhere whether there is a resolution or not.
"We can't define axioms"
I'm not sure what you mean by "define" but what we certainly can do is state axioms in some chosen upon language. I don't know what your issue is with this. It's just some symbols on paper. Here they are.
They're right there, stated with all terms defined. What more do you want? It's written in logic languages and all of the details are out of my expertise but it seems like you may want to look into a textbook about formal languages and see if that addresses your concerns at all. There are all sorts of things trolling around Wikipedia that sound like the things you are talking about.
Sorry to sort of give up but you have all of this set up in your head differently than I have it set up and I think everyone is just talking past eachother at this point since we can't explain our ideas in the other person's paradigm. I want to encourage you to realize that you're probably not the first person to think about this and that there probably exists a thorough discussion of it somewhere whether there is a resolution or not.
Re: Axiomatic mathematics has no foundation
doogly wrote:Treatid wrote:So, we can't define a fixed point. We can't define axioms.
Well, some of us can.
You really can't.
In order to specify a set of axioms you need a language. That language needs to be defined. So you have a language to define the language to define the axioms. Of course you need a language to define the language to define the language that defines the set of axioms.
There is no end point.
For a natural language you may be able to look up the definition of words in a dictionary. In that dictionary, each word is defined by reference to other words. And those other words are defined by reference to yet more words... until you run out of words or start reusing words. In the former case you haven't defined anything, in the latter case you have defined a tautology.
The whole point of axioms is that without a context, a statement has no particular meaning. Axioms provide the context within which we can assess the significance of a statement. But until you have an axiomatic system, statements have no meaning. Then how do you construct your first axiom system? You can't.
There is no starting point from which to build.
This isn't some new insight or bizarre reasoning. We've already established early in the thread that this was the case. Perhaps you might want to take a look at the Foundational Crises in mathematics.
Some take the attitude that while mathematics can't define a fixed point  it does inherit a fixed point from natural language and/or the physics of the universe in which we have evolved. The belief being that natural language can define concepts.
And this is the basis upon which supposed axioms are constructed. But as noted, natural language does not define any words in the axiomatic sense. As much as you think you may understand what particular words and concepts are  you cannot define them without finding yourself in a tautological loop of definitions. A defines B defines C defines A doesn't define anything.
Don't be blinded by what you think you know and perceive. Not having a fixed point upon which to build means that you don't have a fixed point upon which to build. You can't construct axioms (a fixed point) unless you already have a fixed point from which to define those axioms.
Twistar wrote:I was starting to write out another long response but you've lost me.
"We can't define axioms"
I'm not sure what you mean by "define" but what we certainly can do is state axioms in some chosen upon language. I don't know what your issue is with this. It's just some symbols on paper. Here they are.
Some symbols on paper are not a definition. (an undefined language is useless for defining a set of axioms).
The whole point of axioms is that without a definite context there is no way to determine the meaning/significance/definition of a set of symbols. Until you've constructed an axiom system  symbols are meaningless. You can't construct an axiom system until you've constructed an axiom system to define the symbols.
This is the position I started this thread with... and it was greeted by (ahem) polite disagreement back then until korona pointed out that my position was nothing new (Sizik provides a link to the wiki page with good references for various components of the foundational crises).
While I do feel that it is useful for people to understand the implications that arise from the foundational crises... I don't need to defend the fact that it is impossible to define a set of axioms.
The only defence for axiomatic mathematics is the idea that a natural language definition of axioms is not totally meaningless. Axiomatic Mathematics, by itself, has no justification or inherent meaning. Something else must provide both significance and direction. But as much as we do perceive significance and meaning in words... the limitation on defining words without a fixed starting point still exists. As much as you think you know what a word means, the only way to define words using language is by reference to other words... which in turn are defined by reference to other words. If you try to follow the chain of definitions to its ultimate conclusion you will end up where you started  with a tautology (or a contradiction).
It is quite foolish to understand the rigour of an axiomatic system but then forget that rigour when it comes to the language that describes the axiomatic system.
They're right there, stated with all terms defined.
How are those terms defined? By a formal axiomatic system? Many systems can be defined using ZFC as the formal basis... but the language that the axioms of ZFC are specified with is not an axiomatically defined system. Or maybe it is  you can push the rabbit hole back as far as you like. But ultimately you have to admit that there is no way to define the first axiomatic system because you don't have a fixed point upon which to do so. Which means that every subsequent system is also undefined.
What more do you want?
I want mathematicians to stop pretending that they can create a fixed point upon which to build. This is a hard limit on knowledge. There is no circumventing this limit. There is no way to define an axiomatic system.
It's written in logic languages and all of the details are out of my expertise but it seems like you may want to look into a textbook about formal languages and see if that addresses your concerns at all.
Without a preexisting axiom system, you don't have a system of logic. Formal languages are not defined languages.
Sorry to sort of give up but you have all of this set up in your head differently than I have it set up and I think everyone is just talking past eachother at this point since we can't explain our ideas in the other person's paradigm. I want to encourage you to realize that you're probably not the first person to think about this and that there probably exists a thorough discussion of it somewhere whether there is a resolution or not.
I am certainly not the first person to think about this. It has already been established that there is no possible fixed point from which to build an axiomatic system. I am well late to the party in this regard. As such, nobody should be finding it controversial that it isn't possible to specify a set of axioms.
Re: Axiomatic mathematics has no foundation
Mathematicians don't pretend they are creating or have created a fixed point. They take the axioms and formal language within which they work (as undefined as it is) and work with it. That is mathematics. Mathematicians aren't doing anything inconsistent within their framework (the framework I defined in a previous post). Analysists or algebraists are not doing anything inconsistent. They are working within the rules of math and everything is clear. Maybe you take issue with foundational mathematicians or philosophers of mathematics who (by my previous definition of mathematics) work slightly outside of the realm of mathematics. But if this is so you can't say people are doing math wrong. It is formal languages or something else they are doing wrong.
"It is quite foolish to understand the rigour of an axiomatic system but then forget that rigour when it comes to the language that describes the axiomatic system."
No. The book titled MATH is ENTIRELY rigorous except for the first page. That is why I defined math the way I did. It is nicely self contained and rigorous and since it is built so straightforwardly on a small number of nonrigorous statements it is easy to locate and analyze those statements. You think some combination of the following: 1) people think the first page is rigorous and they shouldn't think this. 2) you think the first page should be rigorous for some reason I don't know. Or 3) you think it shouldn't be structured as this book at all.
The thing is if you follow your argument to completion (which you clearly have) you find that humans can't unambiguously communicate anything and you end up in some solipsistic nightmare. Forget getting mathematics back from here, how do we get anything back from here?
I think this is the point in the conversation where you need to give examples of what your new math would look like because you've spent a lot of time explaining things it is impossible for math to get around but you somehow have ideas that would get around it.
"It is quite foolish to understand the rigour of an axiomatic system but then forget that rigour when it comes to the language that describes the axiomatic system."
No. The book titled MATH is ENTIRELY rigorous except for the first page. That is why I defined math the way I did. It is nicely self contained and rigorous and since it is built so straightforwardly on a small number of nonrigorous statements it is easy to locate and analyze those statements. You think some combination of the following: 1) people think the first page is rigorous and they shouldn't think this. 2) you think the first page should be rigorous for some reason I don't know. Or 3) you think it shouldn't be structured as this book at all.
The thing is if you follow your argument to completion (which you clearly have) you find that humans can't unambiguously communicate anything and you end up in some solipsistic nightmare. Forget getting mathematics back from here, how do we get anything back from here?
I think this is the point in the conversation where you need to give examples of what your new math would look like because you've spent a lot of time explaining things it is impossible for math to get around but you somehow have ideas that would get around it.
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Re: Axiomatic mathematics has no foundation
Treatid wrote:Some take the attitude that while mathematics can't define a fixed point  it does inherit a fixed point from natural language and/or the physics of the universe in which we have evolved. The belief being that natural language can define concepts.
And this is the basis upon which supposed axioms are constructed. But as noted, natural language does not define any words in the axiomatic sense. As much as you think you may understand what particular words and concepts are  you cannot define them without finding yourself in a tautological loop of definitions. A defines B defines C defines A doesn't define anything.
Why can axioms only come from other axioms? A mind doesn't run on axioms, and it understands abstract concepts fine  even hypothetical ones. In some sense, the foundation you're looking for is just the conceptual mind. Concepts exist, like we've agreed. The concept of concepts exists, and the concept of concepts that are true in every possible case exists. Therefore the concept of a relationship of concepts that are always true exists, and that's what we build on.
If you use "definition" to mean a product of axioms, then of course there's no foundation. Other people are essentially saying that axioms are defined by their own existance in conceptspace, and the concept of a specific axiom is its definition.
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Re: Axiomatic mathematics has no foundation
Are you (a(n)):
1.) Illogical Cook

2.) Math Crank

3.) A Lingering Post Modernist Who Just Heard About Deconstruction For the 1st time

4.) A Philosophy Undergrad From the 90's Who is Somehow Posting in The Present

5.) A Lover of Blatant Tautologies

6.) Very Arrogant

7.) Physics Crank

8.) Someone angry that math is hard and feels it shouldn't be because they already know something mathlike?
1.) Illogical Cook
I don't need to defend the fact that it is impossible to define a set of axioms.

2.) Math Crank
So, we can't define a fixed point. We can't define axioms. Without axioms there is no logic, no proof. But we already knew that. So why has noone claimed all six of the open Clay Institute Millennial prizes by pointing out that the questions are all undefined? How can anyone pretend that there is some significance to the question of whether P=NP when "P", "=", and "NP" cannot be defined? I can understand accountants, engineers and students going along with the illusion that unicorns axioms exist... but anyone who knows the first thing about axioms should be aware that they can't be stated.

3.) A Lingering Post Modernist Who Just Heard About Deconstruction For the 1st time
As much as you think you know what a word means, the only way to define words using language is by reference to other words... which in turn are defined by reference to other words. If you try to follow the chain of definitions to its ultimate conclusion you will end up where you started  with a tautology (or a contradiction).

4.) A Philosophy Undergrad From the 90's Who is Somehow Posting in The Present
The first step to understanding is realising that there is nospoonaxiomatic mathematics.

5.) A Lover of Blatant Tautologies
Not having a fixed point upon which to build means that you don't have a fixed point upon which to build.

6.) Very Arrogant
I want mathematicians to stop pretending that they can create a fixed point upon which to build. This is a hard limit on knowledge. There is no circumventing this limit. There is no way to define an axiomatic system.

7.) Physics Crank
This is particularly relevant to small scale physics where you might end up describing behaviour in terms of emergent properties that haven't yet emerged at those scales.

8.) Someone angry that math is hard and feels it shouldn't be because they already know something mathlike?
My own experience with programming made it very hard for me to separate the idea of deterministic computer programs and an unambiguous definition of axioms.
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Re: Axiomatic mathematics has no foundation
Treatid wrote: Perhaps you might want to take a look at the Foundational Crises in mathematics.
Yeah let's go take a look at that, cause it got resolved many decades ago.
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Re: Axiomatic mathematics has no foundation
As far as I can tell, all of Treatid's rambling boils down to them being unhappy that God hasn't given us Objective Knowledge, which is undeniably true and factual and existent and intuitively known by all people.
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Re: Axiomatic mathematics has no foundation
Xanthir wrote:As far as I can tell, all of Treatid's rambling boils down to them being unhappy that God hasn't given us Objective Knowledge, which is undeniably true and factual and existent and intuitively known by all people.
Yeah, I think he definitely realizes this and is upset about it. He's specifically upset about how it hurts math in particular so he's trying to find some way around it, but what that way around it might be is supremely unclear. Everyone else's arguments (at least my own) are that he just shouldn't be upset about this.
Re: Axiomatic mathematics has no foundation
Treatid wrote:The only defence for axiomatic mathematics is the idea that a natural language definition of axioms is not totally meaningless. Axiomatic Mathematics, by itself, has no justification or inherent meaning. Something else must provide both significance and direction. But as much as we do perceive significance and meaning in words... the limitation on defining words without a fixed starting point still exists. As much as you think you know what a word means, the only way to define words using language is by reference to other words... which in turn are defined by reference to other words. If you try to follow the chain of definitions to its ultimate conclusion you will end up where you started  with a tautology (or a contradiction).
This whole paragraph but especially the sentence "the only way to define words using language is by reference to other words" strikes me as an extreme hand waving of all of semiotics. Even the long dead forefather of linguistics, Saussure, can refute that easily and there is a multitude of approaches that you should know when you make such a statement. (maybe begin here) Maybe mathematicians will let you get away with throwing all that out the window without any real argument but philosophers and linguists will not.
Please be gracious in judging my english. (I am not a native speaker/writer.)
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Re: Axiomatic mathematics has no foundation
Twistar wrote:Xanthir wrote:As far as I can tell, all of Treatid's rambling boils down to them being unhappy that God hasn't given us Objective Knowledge, which is undeniably true and factual and existent and intuitively known by all people.
Yeah, I think he definitely realizes this and is upset about it. He's specifically upset about how it hurts math in particular so he's trying to find some way around it, but what that way around it might be is supremely unclear. Everyone else's arguments (at least my own) are that he just shouldn't be upset about this.
lorb wrote:Treatid wrote:The only defence for axiomatic mathematics is the idea that a natural language definition of axioms is not totally meaningless. Axiomatic Mathematics, by itself, has no justification or inherent meaning. Something else must provide both significance and direction. But as much as we do perceive significance and meaning in words... the limitation on defining words without a fixed starting point still exists. As much as you think you know what a word means, the only way to define words using language is by reference to other words... which in turn are defined by reference to other words. If you try to follow the chain of definitions to its ultimate conclusion you will end up where you started  with a tautology (or a contradiction).
This whole paragraph but especially the sentence "the only way to define words using language is by reference to other words" strikes me as an extreme hand waving of all of semiotics. Even the long dead forefather of linguistics, Saussure, can refute that easily and there is a multitude of approaches that you should know when you make such a statement. (maybe begin here) Maybe mathematicians will let you get away with throwing all that out the window without any real argument but philosophers and linguists will not.
Yeah.
Having read all three pages of this stuff (more fool me) I still stand by my initial reply: Treadid is expecting maths to be the most fundamental discipline but it really isn't. The question of how Meaning bootstraps itself into existence is a matter of philosophy; Mathematicians simply accept as given that they can write down Meaningful axioms and from them derive Meaningful conclusions  and this irks Treadid for some reason. But really it shouldn't. It works because it works. It's not true to say that 'axiomatic mathematics has no foundation': It has a foundation  just not one within itself.
If you want to look more deeply into that particular mystery then you need to delve into philosophy/linguistics for the answers, not maths.
Re: Axiomatic mathematics has no foundation
Twistar wrote:The thing is if you follow your argument to completion (which you clearly have) you find that humans can't unambiguously communicate anything and you end up in some solipsistic nightmare. Forget getting mathematics back from here, how do we get anything back from here?
Yes. Except:
We don't end up in some solipsistic nightmare. That is the position that existed all along. Humans cannot unambiguously communicate anything.
However, we can reproduce some things with a good degree of fidelity (e.g. computers) (reproduction is not the same as communication). Not being able to communicate unambiguously is undoubtedly inconvenient... but we have still managed to create modern society despite this limitation. Clinging to an illusion because reality seems scary is silly. Reality is what it is  Rejecting reality in favour of an impossible ideal is generally known as delusional.
I think this is the point in the conversation where you need to give examples of what your new math would look like because you've spent a lot of time explaining things it is impossible for math to get around but you somehow have ideas that would get around it.
Brilliant. An opportunity to move forward.
As noted, the first step is to reject the illusion that axioms exist. To accept that we cannot define anything in an axiomatic fashion. This is easier said than does as illustrated by the number of people who know that it is impossible to define a fixed point... and then continue trying to define axioms. Given the several million dollar prizes for axiom based problems... the cognitive dissonance between what mathematicians know, and how they behave illustrates how nontrivial it is to change such deeply held assumptions.
Throwing out the idea of axioms doesn't mean throwing out all of existing mathematics (Though I admit the thought has crossed my mind). Axioms never existed. So existing mathematics is not axiom based. For the most part, mathematics is based on trial and error. Existing systems are valid because they work. The only things lost along with axioms are any justification for mathematics beyond " it works" and the idea that the mathematics as a discipline is understood.
Having ruled out fixed starting points, the only thing we are left with is relationships. All language can communicate is relationships. All we can know is some manifestation of relationships.
This actually makes life easier. We don't have to worry about what "length", "electron", "Spin", "line", "integer" or any other concepts are. We know from the outset that we cannot axiomatically define any object.
We can have an object... we just can't know anything about it. We can't even be certain that two objects are indeed distinct objects  or that a single object is definitively singular. We can't even define existence in a definitive way... as such, even the existence of a given object is somewhat vague.
Similarly, we have relationships between objects. The nature of a relationship is as unknowable as that of any other object. Indeed there is nothing to show that relationships and objects are in any way distinct.
We can't know that relationships are symmetrical... that a>b implies b>a. We shall regard each relationship as directional.
We can regard objects as vertices in a digraph and relationships as edges in a digraph.
At this point there is no justification for objects to be discrete or for relationships to be onetoone. Without a fixed starting point, it isn't going to be possible to justify anything in an absolute manner... and even when justification is possible it is only with respect to some other feature which, itself can only be justified with respect to other features. It is necessary to get used to the idea that everything is a tautology... because there isn't an alternative. However, not all tautologies are the same tautology.
The interesting way to see that a standard digraph is sufficiently representative is to visualise (e.g.) a complete digraph and then extend both the vertices and edges such that they form continuous manifolds.
The more mundane approach is to know that mathematics has already been here. Functions tend to be onetoone and directional, not least because all other combinations can be constructed from this premise. Potentially continuous systems (Euclidean space, Real numbers) are usually treated as sets of discrete points with similar convenience and effect.
Or simply "because it works".
Without knowing anything about objects, or about individual relationships... the only structure we have is the network of relationships we can construct. Patterns of relationships are the basis of everything (not because we chose them as an axiomatic starting point... because we have ruled out every other possibility and this is all we are left with).
We've now established that all communication and all knowledge is represented by the set of all unique digraphs  somehow (because there is nothing else that can be communicated).
But then, the same argument could be made for all sets, for all bit strings, for all Turing Machines (with all inputs) or any other set of symbols (that we could putatively construct).
Axiomatic mathematics went to a lot of trouble to prove beyond any shadow of a doubt that we can't define anything except by reference to other (equally undefinable) things.
"Meaning", "significance", "value" can only exist with respect to something else.
In this instance, the obvious "something else" that we are interested in is ourselves. The observer is an intimate component of mathematics. This specifically means that both the observer and subject must exist within the same system. There is no definable (fixed) external perspective for an observer. The observer is an essential component of the system being observed.
Moreover, each nonidentical observer in a system necessarily has a different conception of meaning to every other observer AND different observations with respect to the system they inhabit. The degree of difference in cognition and perception between a given two observers is potentially quantifiable and an approximation of mapping can be constructed between the two.
We might call this mapping something like "empathy", perhaps.
With an explicit observer as part of the system we at least have the beginnings of a reference frame (but not an axiomatic fixed point). With this and an understanding of emergence we have a basis for understanding mathematics. Note (again) that mathematics has already been constrained to work within this system. We have always been the observer within the systems we observe (whether implicit or explicit). Axioms have never existed. We have only ever been able to communicate networks of relationships. The fault of mathematics is not what it has done... but what it thinks it is trying to do.
Twistar wrote:Yeah, I think he definitely realizes this and is upset about it.
I'm not upset that there are hard limits on knowledge. My frustration, such as it is, derives from mathematic's (or mathematician's) apparent stubbornness in the face of these limits. I feel that mathematics should embrace its own results, yet the impression I get is that this particular result is troublesome so we'll pretend it doesn't exist. Hence we have a situation where mathematicians do know that axioms can't exist, yet are pretending they do even to the point of offering million dollar prizes for problems that cannot be defined.
To know something and yet act contrary to that knowledge is delusional. The emperor IS naked.
It is a good argument to say that existing mathematics works. I'm completely onboard with that argument. What I cannot accept is the delusion that axioms are any part of what makes mathematics work.
Xanthir wrote:As far as I can tell, all of Treatid's rambling boils down to them being unhappy that God hasn't given us Objective Knowledge, which is undeniably true and factual and existent and intuitively known by all people.
Again  no. I'm perfectly happy that we don't have objective knowledge. What bothers me is that despite all protestations to the contrary, mathematics clearly has not embraced this. (which tends to put the lie to your assertion that people "intuitively understand there is no objective knowledge". Right in this thread you can find plenty of people arguing that they can define a set of axioms).
When pushed, people are conceding that there can be no objective truth. But mathematics doesn't behave according to this understanding. Axiomatic systems such as set theories, category theories, group theories and all the rest are still treated as meaningful entities with inherent truths. The hierarchy of infinities are defined with respect to a fixed point that doesn't exist and mathematicians nod their heads sagely. Proof is regarded as the bedrock of mathematics... despite mathematicians supposedly knowing that there is no such thing as proof, no way to define 'true' and 'false'.
I could understand a mathematician saying "yes  this is a huge problem for axiomatic mathematics... we just don't see any alternative". But instead, the strong impression I am getting is "there is a small technical issue but it isn't of any relevance to actual axiomatic mathematics". I simply can't get my head around the idea that the complete nonexistence of axioms is a "small technical issue" for axiomatic mathematics.
I expect to find people following the status quo in religions despite evidence. To see exactly the same characteristic among mathematicians is disappointing.
This isn't an emergency. Your house isn't burning down (as far as I know). Economies aren't going to be destroyed and it has no impact on Russia annexing Crimea.
But for anything connected to axiomatic mathematics, the nonexistence of axioms is a big deal. It doesn't matter if it was also a big deal fifty or a hundred years ago too. Whether for the field of mathematics or you personally, the age of the information is of no relevance. It will always be a big deal for axiomatic mathematics.
So, help me out... you know there is no objective knowledge. You know that the axioms of ZFC aren't axioms. You know that the components of P=NP cannot be defined. You know that none of the Millennial Prize problems can be defined. Yet mathematicians still talk about ZFC as a real thing; The Millennial Prizes still exist. How does this make sense to you? Explain to me why the nonexistence of axioms is largely irrelevant to axiomatic theory, please.
doogly wrote:Treatid wrote: Perhaps you might want to take a look at the Foundational Crises in mathematics.
Yeah let's go take a look at that, cause it got resolved many decades ago.
You have no idea what you are talking about.
http://en.wikipedia.org/wiki/Foundations_of_mathematics
Magnanimous wrote:Why can axioms only come from other axioms? A mind doesn't run on axioms, and it understands abstract concepts fine  even hypothetical ones. In some sense, the foundation you're looking for is just the conceptual mind. Concepts exist, like we've agreed. The concept of concepts exists, and the concept of concepts that are true in every possible case exists. Therefore the concept of a relationship of concepts that are always true exists, and that's what we build on.
If you use "definition" to mean a product of axioms, then of course there's no foundation. Other people are essentially saying that axioms are defined by their own existance in conceptspace, and the concept of a specific axiom is its definition.
That the mind exists and holds, manipulates and constructs concepts is an important observation. We do perceive meaning and significance. What we don't do is to perceive anything in an absolute sense.
We can imagine things that don't exist, like unicorns.
Conceiving of outright impossible things we can't do. "An unstoppable force meets and immovable object". For the most part we can imagine the component elements of this particular impossibility. We can see where the impossibility exists... but we can't resolve the impossibility itself.
Similarly we can know what axioms are as a general idea. But we can't actually imagine an axiom. Whether implicitly or explicitly, the structure of axioms does assume the existence of other axioms. Specifically, in order to construct a fixed point we first need a fixed point. And axioms are the only source of fixed points.
It isn't unreasonable to suppose that there did exist some fixed point independent of axioms upon which everything else could be built. I would argue that a significant portion of mathematics is an attempt to find or establish such a fixed point. Mathematics has demonstrated conclusively that a fixed point cannot be constructed using the tools of mathematics. Potentially that does still leave the possibility that the universe has gifted us with a fixed point.
Unfortunately (in this instance) mathematics is able to generalise some results. The results that show mathematics can't create a fixed point apply to all the system we can conceive of. Unless you decide that humans and this universe are not 'conceivable systems'... human imagination is constrained in the same way that mathematics is. So... either humans and this universe are inconceivable entities that we can never understand... or there are no fixed points from which to construct axioms.
elasto wrote:Yeah.
Having read all three pages of this stuff (more fool me) I still stand by my initial reply: Treadid is expecting maths to be the most fundamental discipline but it really isn't. The question of how Meaning bootstraps itself into existence is a matter of philosophy; Mathematicians simply accept as given that they can write down Meaningful axioms and from them derive Meaningful conclusions  and this irks Treadid for some reason. But really it shouldn't. It works because it works. It's not true to say that 'axiomatic mathematics has no foundation': It has a foundation  just not one within itself.
If you want to look more deeply into that particular mystery then you need to delve into philosophy/linguistics for the answers, not maths.
I agree with much of what you say. I do regard mathematics as the most formal of the sciences, the most selfcontained. There certainly isn't anything more qualified for that role.
I also agree that there is more than a hint of philosophy and linguistics.
However "Mathematicians simply accept as given that they can write down Meaningful axioms and from them derive Meaningful conclusions" is causing me to choke. As much as mathematicians may believe that this is the case... it simply isn't true.
It isn't possible to state a set of axioms. Whether leaning on external systems or not, there is no way to specify a set of axioms. External systems do not provide the fixed point that is required.
Yes  mathematics works. We've established that trial and error will produce useful results. But the fact that a system works is not evidence that the system is well formed.
I see your argument that this belongs to philosophy or linguistics... And I don't agree. You are, in effect, arguing that Godel's theorems don't belong in mathematics. Self examination is very much part of mathematics.
Re: Axiomatic mathematics has no foundation
You do a disservice to how ethereal mathematical axioms really are. You say: "As noted, the first step is to reject the illusion that axioms exist"
Then you say: "Without knowing anything about objects, or about individual relationships... the only structure we have is the network of relationships we can construct. Patterns of relationships are the basis of everything"
Which is how axioms are defined!
eg. Set theory says "We define a 'set' to be 'something which can contain something'". Note that it doesn't even need to define what 'contains' really means! All we are really assuming here is that it's possible for 'something' to exist and that it's possible for 'things to have relationships'.
Then it says 'assume there exists a set which doesn't contain anything'. This has to be its own assumption because it's possible that there's nothing that doesn't have a relationship with something (perhaps itself) for a given definition of 'contains'.
So we haven't defined what the 'something' is  only that it's possible for it to exist. We haven't assumed what the 'relationship' is  only that it exists. Yet out of that (and a few other things) a rich tapestry of interesting and complicated results follow.
No I'm not. You're assuming there is some sharp boundary between mathematics and philosophy  but the edges are blurred just as they are between biology and physics or any other field of study. Again, I said this in my very first post.
But I *am* saying that a lot of the stuff you're getting into, like 'words can have no meaning because they are only defined in terms of other words' and hence 'axioms can have no meaning because they are defined in terms of words which we have already proved have no meaning' are much better dealt with using the tools of philosophy (or linguistics  again, the boundaries are blurred) than the tools of mathematics.
Use the right tool for the right job and your life will become much simpler!
Then you say: "Without knowing anything about objects, or about individual relationships... the only structure we have is the network of relationships we can construct. Patterns of relationships are the basis of everything"
Which is how axioms are defined!
eg. Set theory says "We define a 'set' to be 'something which can contain something'". Note that it doesn't even need to define what 'contains' really means! All we are really assuming here is that it's possible for 'something' to exist and that it's possible for 'things to have relationships'.
Then it says 'assume there exists a set which doesn't contain anything'. This has to be its own assumption because it's possible that there's nothing that doesn't have a relationship with something (perhaps itself) for a given definition of 'contains'.
So we haven't defined what the 'something' is  only that it's possible for it to exist. We haven't assumed what the 'relationship' is  only that it exists. Yet out of that (and a few other things) a rich tapestry of interesting and complicated results follow.
I see your argument that this belongs to philosophy or linguistics... And I don't agree. You are, in effect, arguing that Godel's theorems don't belong in mathematics. Self examination is very much part of mathematics.
No I'm not. You're assuming there is some sharp boundary between mathematics and philosophy  but the edges are blurred just as they are between biology and physics or any other field of study. Again, I said this in my very first post.
But I *am* saying that a lot of the stuff you're getting into, like 'words can have no meaning because they are only defined in terms of other words' and hence 'axioms can have no meaning because they are defined in terms of words which we have already proved have no meaning' are much better dealt with using the tools of philosophy (or linguistics  again, the boundaries are blurred) than the tools of mathematics.
Use the right tool for the right job and your life will become much simpler!
 Schrollini
 Posts: 515
 Joined: Sat Sep 29, 2012 5:20 pm UTC
Re: Axiomatic mathematics has no foundation
elasto already said it, but I'm going to repeat it:
These things are called sets.
These things are called axioms.
Congratulations, you've invented set theory.
No, really, you're trying to implement some set theory, but with different words for everything. Set theory starts with these objects that we don't know anything about, and whose existence is somewhat vague. We define a basic relationship called "contained in". Other axioms use this relationship to build more complicated relationships. For example, we say that two sets x and y are equal if, for every third set z, z is contained in x if and only if z is contained in y. Voila, the axiom of extensionality.
I'm sure you're going to object to this characterization. And I'm sure you'll do it with your own vocabulary, to make it more difficult for the rest of us to counter. Have fun with that. In the meantime, I'm going to be snarky:
Hey everyone, let's play MathLibs!
To get you started:
participles: extending, reducing, removing, integrating, substituting
adjectives: continuous, piecewise, complete, differentiable, divergent
adverbs: monotonically, logarithmically, centrally, smoothly, sufficiently
nouns: group, symmetry, manifold, representation, axiom
You, on the other hand, learned about this a whole month ago, and therefore understand it perfectly.
Treatid wrote:We can have an object... we just can't know anything about it. We can't even be certain that two objects are indeed distinct objects  or that a single object is definitively singular. We can't even define existence in a definitive way... as such, even the existence of a given object is somewhat vague.
These things are called sets.
Treatid wrote:Similarly, we have relationships between objects. The nature of a relationship is as unknowable as that of any other object. Indeed there is nothing to show that relationships and objects are in any way distinct.
These things are called axioms.
Congratulations, you've invented set theory.
No, really, you're trying to implement some set theory, but with different words for everything. Set theory starts with these objects that we don't know anything about, and whose existence is somewhat vague. We define a basic relationship called "contained in". Other axioms use this relationship to build more complicated relationships. For example, we say that two sets x and y are equal if, for every third set z, z is contained in x if and only if z is contained in y. Voila, the axiom of extensionality.
I'm sure you're going to object to this characterization. And I'm sure you'll do it with your own vocabulary, to make it more difficult for the rest of us to counter. Have fun with that. In the meantime, I'm going to be snarky:
Treatid wrote:The interesting way to see that a standard digraph is sufficiently representative is to visualise (e.g.) a complete digraph and then extend both the vertices and edges such that they form continuous manifolds.
Hey everyone, let's play MathLibs!
By (participle) the (adjective) (noun), we can show that the (noun) converges (adverb) to the (adjective) solution, as predicted by (mathematician)
To get you started:
participles: extending, reducing, removing, integrating, substituting
adjectives: continuous, piecewise, complete, differentiable, divergent
adverbs: monotonically, logarithmically, centrally, smoothly, sufficiently
nouns: group, symmetry, manifold, representation, axiom
Treatid wrote:doogly wrote:Treatid wrote: Perhaps you might want to take a look at the Foundational Crises in mathematics.
Yeah let's go take a look at that, cause it got resolved many decades ago.
You have no idea what you are talking about.
You, on the other hand, learned about this a whole month ago, and therefore understand it perfectly.
For your convenience: a LaTeX to BBCode converter
 Forest Goose
 Posts: 377
 Joined: Sat May 18, 2013 9:27 am UTC
Re: Axiomatic mathematics has no foundation
@Treatid:
Your position looks, to me, to be an averaging of views expressed in:
http://www.eduneg.net/generaciondeteoria/files/Lakatos%201976%20A%20renaissance%20of%20empiricism%20in%20the%20recent%20philosophy%20of%20mathematics.pdf
http://www.helsinki.fi/~pietarin/publications/PragmaticismAntifoundationalismPietarinen.pdf
http://www.iep.utm.edu/mathfict/
http://plato.stanford.edu/entries/fictionalismmathematics/
http://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma
,But arrived at by you, in your own special way, drawing your own neat conclusions.
Would this be an accurate representation of your views? Do any of the links approximate what you are trying to say?
Do you agree with any of the ideas in the criticisms sections of:
http://en.wikipedia.org/wiki/Foundationalism#Criticisms
http://en.wikipedia.org/wiki/Correspondence_theory_of_truth#Objections?
My biggest problem with everything you are saying is that:
1.) You do not know advanced mathematics
2.) You repurpose already defined words
3.) You aren't providing arguments, just stating conclusions; over and over and over
4.) A while ago you, in past threads, were proving all the major problems of science and mathematics; after being shown wrong, you are now insisting that math just doesn't work  it makes you seem more like you are bitter about that than you actually have a point.
Essentially, your thought process seems to have evolved (decayed?) as:
1.) I like math and physics
2.) These unsolved problems are neat
3.) I have an insight
4.) I solved them
5.) I solved them, but haven't filled in the details (but I'm right, I'm a big picture kind of guy...)
6.) I can't fill in the details
7.) The details must not make sense
8.) I should be right, I'm not, it must all be illfounded
9.) I know the real foundation,
I don't like this, it insults everyone who has invested countless hours learning something on the basis that you can't wrap the subject up with only a superficial glance. I await:
10.) Using my real foundation: I'm now right and all who disagree are wrong.
Especially because of sentiment such as:
Where you place yourself above the dishonest mathematicians; portraying yourself as the lone sane man holding the candle to light the way.
Also  given that you attempted to prove P=NP earlier, but had no idea what you were saying 
,And other such related to Millennium problems (and physics), seem to suggest that you are upset that your "version" isn't right; and that you, thus, intend to argue philosophy till it becomes so (, which it won't).
All in all, it seems disingenuous as opposed to ingenious.
Your position looks, to me, to be an averaging of views expressed in:
http://www.eduneg.net/generaciondeteoria/files/Lakatos%201976%20A%20renaissance%20of%20empiricism%20in%20the%20recent%20philosophy%20of%20mathematics.pdf
http://www.helsinki.fi/~pietarin/publications/PragmaticismAntifoundationalismPietarinen.pdf
http://www.iep.utm.edu/mathfict/
http://plato.stanford.edu/entries/fictionalismmathematics/
http://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma
,But arrived at by you, in your own special way, drawing your own neat conclusions.
Would this be an accurate representation of your views? Do any of the links approximate what you are trying to say?
Do you agree with any of the ideas in the criticisms sections of:
http://en.wikipedia.org/wiki/Foundationalism#Criticisms
http://en.wikipedia.org/wiki/Correspondence_theory_of_truth#Objections?
My biggest problem with everything you are saying is that:
1.) You do not know advanced mathematics
2.) You repurpose already defined words
3.) You aren't providing arguments, just stating conclusions; over and over and over
4.) A while ago you, in past threads, were proving all the major problems of science and mathematics; after being shown wrong, you are now insisting that math just doesn't work  it makes you seem more like you are bitter about that than you actually have a point.
Essentially, your thought process seems to have evolved (decayed?) as:
1.) I like math and physics
2.) These unsolved problems are neat
3.) I have an insight
4.) I solved them
5.) I solved them, but haven't filled in the details (but I'm right, I'm a big picture kind of guy...)
6.) I can't fill in the details
7.) The details must not make sense
8.) I should be right, I'm not, it must all be illfounded
9.) I know the real foundation,
I don't like this, it insults everyone who has invested countless hours learning something on the basis that you can't wrap the subject up with only a superficial glance. I await:
10.) Using my real foundation: I'm now right and all who disagree are wrong.
Especially because of sentiment such as:
My frustration, such as it is, derives from mathematic's (or mathematician's) apparent stubbornness in the face of these limits. I feel that mathematics should embrace its own results, yet the impression I get is that this particular result is troublesome so we'll pretend it doesn't exist. Hence we have a situation where mathematicians do know that axioms can't exist, yet are pretending they do even to the point of offering million dollar prizes for problems that cannot be defined.
It is a good argument to say that existing mathematics works. I'm completely onboard with that argument. What I cannot accept is the delusion that axioms are any part of what makes mathematics work.
The hierarchy of infinities are defined with respect to a fixed point that doesn't exist and mathematicians nod their heads sagely. Proof is regarded as the bedrock of mathematics... despite mathematicians supposedly knowing that there is no such thing as proof, no way to define 'true' and 'false'.
I expect to find people following the status quo in religions despite evidence. To see exactly the same characteristic among mathematicians is disappointing.
Throwing out the idea of axioms doesn't mean throwing out all of existing mathematics (Though I admit the thought has crossed my mind).
Where you place yourself above the dishonest mathematicians; portraying yourself as the lone sane man holding the candle to light the way.
Also  given that you attempted to prove P=NP earlier, but had no idea what you were saying 
You know that the components of P=NP cannot be defined. You know that none of the Millennial Prize problems can be defined.
,And other such related to Millennium problems (and physics), seem to suggest that you are upset that your "version" isn't right; and that you, thus, intend to argue philosophy till it becomes so (, which it won't).
All in all, it seems disingenuous as opposed to ingenious.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
Re: Axiomatic mathematics has no foundation
elasto wrote:You do a disservice to how ethereal mathematical axioms really are. You say: "As noted, the first step is to reject the illusion that axioms exist"
Then you say: "Without knowing anything about objects, or about individual relationships... the only structure we have is the network of relationships we can construct. Patterns of relationships are the basis of everything"
Which is how axioms are defined!
We have two distinct things:
i) The concept of axioms.
ii) What mathematics does when it claims to be defining axioms.
Defining a set of axioms is impossible.
Whatever mathematics does... it isn't defining sets of axioms.
There is absolutely similarity between what mathematics actually does and what I'm proposing. There are two major differences:
a) Mathematics claims to be doing something impossible (defining axioms).
b) Mathematics tends to assume an actual impossible fixed point (this is distinct from merely claiming to have axioms).
So... the other way around.
The definition of an axiom is a fixed point. We know that those can't be constructed. When mathematicians think they are specifying axioms... they are doing something  but they are not specifying axioms. And since the only thing there is to construct is patterns of relationships... yes, there is a strong similarity between what mathematicians actually do and what I was describing.
It is also true that the various systems that mathematicians describe can be viewed as a network of relationships.
So, to this point, what mathematicians do is much the same as what I described. Which is understandable since there is nothing else that they could conceivably do.
They can't define axioms. They can't create a system of logic. They can't prove anything. The only thing that it is possible to do is to create patterns of relationships.
So yes, take away the intention and delusion... and mathematics already does what it is possible to do.
All I have done is to state explicitly what is possible and what is not possible. Regard it as something similar to the fifth Euclidean postulate. Even without that as an explicit postulate, mathematicians were still doing Euclidean Geometry. Making the fifth postulate explicit didn't significantly change Euclidean Geometry. Mathematicians were already doing Euclidean Geometry even though they hadn't fully understood what they were doing (had they fully understood they would have understood the necessity for the fifth postulate). On the other hand, gaining a more complete understanding such that the requirement for the fifth postulate was clear revealed noneuclidean Geometry.
You are correct that I'm not introducing anything that hasn't already existed. That an observer has always been an implicit part of the process of observation is of no surprise when it is pointed out. That everything we perceive and understand is an emergent feature is barely more of a stretch... certainly emergent properties aren't a surprise in themselves.
Technically, the nonexistence of axioms isn't news either.
And yet mathematicians still talk about proof, about defining axioms, logic,... most damning of all, they still propose that things like empty sets exist and can be taken as a fixed point upon which to construct meaning.
The difference is not in what can be done. It is impossible to do what is impossible. Whatever mathematics does must be what is doable.
The difference lies in understanding what is being done.
Take an internal combustion engine. Strip it down into its component parts. Now present those parts to two people and ask them to put the pieces together. The first person knows nothing about engines, indeed this person has an unshakeable belief that he is constructing a tree. The second person has a good understanding of the principles of internal combustion.
Given time, and sufficient replacement parts, there is no doubt that the first person will eventually be able to put the parts together as a working engine. Trial and error does work. Although faith has been known to lead people to disregard the evidence of their own experiments.
It isn't unreasonable to expect that the second person will have the engine up and running significantly more quickly than the first person.
eg. Set theory says "We define a 'set' to be 'something which can contain something'". Note that it doesn't even need to define what 'contains' really means! All we are really assuming here is that it's possible for 'something' to exist and that it's possible for 'things to have relationships'.
This is fine. If sets (and "contains") are the only components of set theory then I can get over my pedantic insistence that it shouldn't be called axiomatic.
Then it says 'assume there exists a set which doesn't contain anything'. This has to be its own assumption because it's possible that there's nothing that doesn't have a relationship with something (perhaps itself) for a given definition of 'contains'.
And this isn't remotely okay. It is impossible to define an "empty set".
Let us assume that "the empty set" isn't a fixed point. If the empty set isn't a fixed point then we can substitute the empty set for any other set. So... instead of the empty set we have an arbitrary set that contains other sets. We don't know which other sets or even how many other sets.
Now... can we construct number systems using this modified set theory? No.
In fact, I'm pretty sure that there are no axiomatic systems that can be constructed in the conventional way if set theory doesn't include some analogue to "the empty set".
If you are constructing anything using the assumptions of axioms... then you have assumed a fixed point. "If, Then, Else" and "A therefore B" rely on establishing the first condition. That is impossible to do.
I agree with you that set theory has tried very hard to minimise the assumption of fixed points. But even a fraction of a fixed point is still an impossibility. There is no grey area. There is no "little bit pregnant". You are either assuming an (impossible) fixed point or you aren't.
Sets themselves are a reasonable starting point. I'd argue that they are a somewhat indirect representation of relationships... but I can live with mere inefficiency.
Unfortunately, the mechanism of axiomatic systems cannot do anything with just sets. Axiomatic theory doesn't know how to represent emergent properties. As such, some sets with no specific starting point, and no order, provide nothing for an axiomatic system to work with.
So we haven't defined what the 'something' is  only that it's possible for it to exist. We haven't assumed what the 'relationship' is  only that it exists. Yet out of that (and a few other things) a rich tapestry of interesting and complicated results follow.
Yes you have... you have defined that this particular set is empty. That this set has a definite quality that distinguishes uniquely. That we know a specific, definite thing about this set.
That rich tapestry is built upon an impossibility. In fact, that rich tapestry is entirely wish fulfillment. There is no logic that follows from an impossible assumption. Mathematics is the ultimate case of Freudian projection.
That is not to say that mathematics has no value. Again... mathematics has no choice. The systems we build (and find interesting) are ones it is possible to build. There is no justification other than what we bring. There really is no objective truth against which we can evaluate anything. I'm not saying that there is some alternative which is inherently more meaningful.
What there is, is an understanding that is in line with the constraints. See Internal combustion illustration.
No I'm not. You're assuming there is some sharp boundary between mathematics and philosophy  but the edges are blurred just as they are between biology and physics or any other field of study. Again, I said this in my very first post.
Ah  I thought you were suggesting a sharp boundary by claiming that this belonged in philosophy or linguistics but not mathematics.
As it is, I think we agree... the limits of knowledge smacks of philosophy, communication is clearly related to linguistics... and mathematics overlaps with both in defining those limits and looking for ways to express concepts.
But I *am* saying that a lot of the stuff you're getting into, like 'words can have no meaning because they are only defined in terms of other words' and hence 'axioms can have no meaning because they are defined in terms of words which we have already proved have no meaning' are much better dealt with using the tools of philosophy (or linguistics  again, the boundaries are blurred) than the tools of mathematics.
If the conventional tools of philosophy and linguistics were sufficient then those disciplines would be much further advanced than they currently are. On the other hand, a similar argument could be made about mathematics.
I perceive that mathematics has tried to tackle both philosophical and linguistic issues. Mathematics has provided a formal underpinning for solipsism. Mathematics has rigorously defined limits on knowledge and communication. It is mathematics that lets us know that we cannot define a set of axioms. It seems to me that mathematics is still the best placed discipline to follow through on these results. The habits of mathematics... taking an idea and exploring that idea to its limits, looking for inconsistencies and being prepared to accept results despite prejudices if that is where the path leads... are the ideals that lead me to think that mathematics is the right place to pursue this particular avenue.
Use the right tool for the right job and your life will become much simpler!
I agree. And I think the right tool is Relativistic mathematics.
Schrollini wrote:These things are called sets.
I mostly agree. In practice, it has been decided that a set can't contain itself... (at least in some set theories) which is one of those impossible fixed points. And I feel that sets themselves are a bit inefficient. But as a generic object that represents relationships then sets are absolutely already doing what I want. If someone constructs a set theory with just sets, and no fixed points, then I will happily embrace that theory... even if they insist on calling it an axiomatic theory.
The trouble is that set theories don't just contain sets. They contain a special set too.
These things are called axioms.
You can label them whatever you like...but they are not, in fact, axioms.
A label is just a label. If you want to label something as an axiom I can live with that... provided it is fully understood that the object so labelled is not, in any way similar to an axiom.
Congratulations, you've invented set theory.
Gosh... that means I'm smart. Because set theory hasn't been invented yet.
But seriously... set theory hasn't been invented:
1) Anything claiming to be a set of axioms is lying.
2) Anything that appears to be constructed from a set of axioms is mistaken.
3) Anything constructed on an impossibility is itself an impossibility.
Whatever you think that ZFC (and all variants) are... it isn't an axiomatic theory. The results of these systems are whatever we want the results to be... there is no logical deduction. That these systems work for us is purely a reflection of us selecting the systems that work for us. And they do work... but they are not axiomatic systems. They are not well defined theories. They are not logical deductions. They are just things that we find useful.
As far as a digraph having a roughly equivalent representation in sets, and in binary digits, and in Turing Machines and in a host of other symbol systems... I pointed that out myself. All languages describe relationships. Languages don't do anything else. All languages are equivalent in this regard (assuming the language is sufficiently flexible to be a Universal Turing Machine Equivalent and not just a Turing Machine Equivalent).
I then went on to point out that enumerating all the possible statements doesn't tell us anything about those statements. This also applies to all languages, including set theories.
We must explicitly include the observer within the system that is being observed. Here you could argue that humans do exist within the same system as the systems that mathematics is interested in... and you would be right. And I would again tell you that I have already said that mathematics already operates under the constraints that I'm describing... it simply doesn't have a choice.
"Well then, " you argue, "since this is already how things are done, why make such a fuss?"
"Because", I reply, "understanding how something works gives you much more control over what you can do. Trial and error works, but it is inefficient. Correctly understanding the structure of a thing significantly multiplies your potential effectiveness with that thing. As it stands, mathematicians are bumbling their way through using concepts that they know to be flat out impossible. They are fighting themselves every step of the way by clinging to concepts that have been shown can never exist."
No, really, you're trying to implement some set theory, but with different words for everything. Set theory starts with these objects that we don't know anything about, and whose existence is somewhat vague. We define a basic relationship called "contained in". Other axioms use this relationship to build more complicated relationships. For example, we say that two sets x and y are equal if, for every third set z, z is contained in x if and only if z is contained in y. Voila, the axiom of extensionality.
Set theory starts with sets you don't know anything about... except for that special set... that empty set... that set that we know doesn't contain any other sets.
I agree with you about sets as an abstract. You are absolutely right in seeing an equivalence between digraphs and sets.
But you can't establish equivalence (referring to axiom of extensionality) without reference to some absolute. You can't establish order without being able to specify a start and an end.
As much as sets themselves are a generic object... there is a reason why axiomatic mathematics requires the empty set as a special case. Without a fixed starting point, axiomatic mathematics can't do anything. At all.
Combining generic objects with a fixed starting point doesn't solve the fact that you are assuming a fixed starting point... which we know is totally, utterly, and in all ways, impossible.
You, on the other hand, learned about this a whole month ago, and therefore understand it perfectly.
There is nothing wrong with being ignorant or wrong. I was not slapping doogly for being mistaken. It happens to the best of us.
As much as you find me arrogant, wrong, infuriating, stubborn, selfimportant,... You still present arguments. You give reasons for your position. For the most part you take note of the arguments I'm making even as you are disagreeing with those arguments.
That was doogly's third comment in a row that consisted solely of a flat statement with no justification or support and no evidence that he had taken note of anything either I or other interlocutors had discussed. He wasn't trying to educate me and he wasn't interested in any sort of discussion. Which leaves him being a troll  only interested in scoring points.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5362
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Somerville, MA
 Contact:
Re: Axiomatic mathematics has no foundation
Oh honey my patience was done months ago, you can only hope to engage with people who are late to the party at this point. Also with Schrollini, who has the patience of a saint. If Waterman didn't wear him out, you should be fine.
But, look:
The fifth postulate was extremely explicit. That is why it is called the fifth postulate; in Euclid's Elements, it is the fifth one listed. It was right there in the text.
And let's focus then. You are saying that the statement, "An empty set exists," is problematic. It can't be used in any foundational way. Why is that? Do you think there is a communication barrier? If Schrollini says "empty set," that without some angel to whisper truths in my ear, I cannot figure out what the "meaning" of empty set is?
But, look:
treatid wrote:Even without that as an explicit postulate, mathematicians were still doing Euclidean Geometry. Making the fifth postulate explicit didn't significantly change Euclidean Geometry.
The fifth postulate was extremely explicit. That is why it is called the fifth postulate; in Euclid's Elements, it is the fifth one listed. It was right there in the text.
And let's focus then. You are saying that the statement, "An empty set exists," is problematic. It can't be used in any foundational way. Why is that? Do you think there is a communication barrier? If Schrollini says "empty set," that without some angel to whisper truths in my ear, I cannot figure out what the "meaning" of empty set is?
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Axiomatic mathematics has no foundation
The empty set is not some special set. It is just some set that has certain relationships with other sets (objects in your language). The empty set is defined as such:
There exists a set, O (the empty set), such that for every set X, X is not contained in O.
The empty set is only defined by its relationship to other "objects" and you claim we are allowed to talk about relationships.
There exists a set, O (the empty set), such that for every set X, X is not contained in O.
The empty set is only defined by its relationship to other "objects" and you claim we are allowed to talk about relationships.
Re: Axiomatic mathematics has no foundation
I can't tell, are you rejecting the idea of talking about a thing, and then giving that thing a name?
If not, then when someone says "the empty set", just think "a/the set that doesn't have the contain relation with anything"
because its just a name.
And of course, replacing the name with the desc should pretty much yield the same result, just in a stupidly long and hard to read manner, so rejecting names doesn't make any sense.
You might say/argue/state that giving a thing a name suggests "uniqueness" or something, and that that would be misleading or something,
but in pretty much all cases (I have not though of any counterexamples)
calling two things that have the same properties/relations , and as such cannot be distinguished, "the same thing" doesn't have any bad results.
So then, patching/clarifying that "hole" in my argument, I again conclude that denying giving things names is not particularly useful.
Although if you want to call it something other than "empty" because you believe that the word has some thing that changes meaning or something, you could do that without issue, other than that it would hinder communication.
:emoticon:
oh also, do you have any thoughts about tautologies, and general implication stuff?
like do you accept a>a ?
and a V !a
etc.
If not, then when someone says "the empty set", just think "a/the set that doesn't have the contain relation with anything"
because its just a name.
And of course, replacing the name with the desc should pretty much yield the same result, just in a stupidly long and hard to read manner, so rejecting names doesn't make any sense.
You might say/argue/state that giving a thing a name suggests "uniqueness" or something, and that that would be misleading or something,
but in pretty much all cases (I have not though of any counterexamples)
calling two things that have the same properties/relations , and as such cannot be distinguished, "the same thing" doesn't have any bad results.
So then, patching/clarifying that "hole" in my argument, I again conclude that denying giving things names is not particularly useful.
Although if you want to call it something other than "empty" because you believe that the word has some thing that changes meaning or something, you could do that without issue, other than that it would hinder communication.
:emoticon:
oh also, do you have any thoughts about tautologies, and general implication stuff?
like do you accept a>a ?
and a V !a
etc.
I found my old forum signature to be awkward, so I'm changing it to this until I pick a better one.
 Schrollini
 Posts: 515
 Joined: Sat Sep 29, 2012 5:20 pm UTC
Re: Axiomatic mathematics has no foundation
Treatid wrote:We have two distinct things:
i) The concept of axioms.
ii) What mathematics does when it claims to be defining axioms.
No, the two things are what mathematics thinks axioms are and what you think axioms are. (After all, you've just proven that there are no fixed meanings.) What mathematics thinks axioms are is what axioms are for mathematics. (You like tautologies, right?) When we say "axiom", we mean thingmathematicsmeansbyaxiom. If you mean something else by axiom, you're wrong.
Note that the mathematical definition of axiom may be different from the popular definition. That's okay. You wouldn't say that the mathematical definition of a group is wrong because a group of people doesn't have an identity element.
Treatid wrote:All I have done is to state explicitly what is possible and what is not possible. Regard it as something similar to the fifth Euclidean postulate. Even without that as an explicit postulate, mathematicians were still doing Euclidean Geometry. Making the fifth postulate explicit didn't significantly change Euclidean Geometry. Mathematicians were already doing Euclidean Geometry even though they hadn't fully understood what they were doing (had they fully understood they would have understood the necessity for the fifth postulate). On the other hand, gaining a more complete understanding such that the requirement for the fifth postulate was clear revealed noneuclidean Geometry.
doogly has already pointed this out, but you've missed it once already, so I'll repeat: Euclid's fifth axiom has been around since Euclid (or maybe before). As long as mathematicians have been doing Euclidean geometry, they've been doing so with an explicit fifth axiom.
Treatid wrote:Schrollini wrote:These things are called sets.
I mostly agree. In practice, it has been decided that a set can't contain itself... (at least in some set theories) which is one of those impossible fixed points. And I feel that sets themselves are a bit inefficient. But as a generic object that represents relationships then sets are absolutely already doing what I want. If someone constructs a set theory with just sets, and no fixed points, then I will happily embrace that theory... even if they insist on calling it an axiomatic theory.
Care to point me to the axiom in ZF that says sets can't be elements of themselves? I can't find it.
ZF avoids Russel's paradox not by strengthening axioms, but weakening them. Specifically, it replaces the axiom of unrestricted comprehension of naive set theory with the axiom schema of specification. It doesn't say the Russel's paradoxical set doesn't exist; it just removes the guarantee that it does. I would have though that you'd approve of such an approach.
Treatid wrote:Set theory starts with sets you don't know anything about... except for that special set... that empty set... that set that we know doesn't contain any other sets.
As I pointed out before, there are formulations of ZF that don't contain an axiom of the empty set. Instead, it is derived from the axiom schema of specification.
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 Forest Goose
 Posts: 377
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Re: Axiomatic mathematics has no foundation
I'm having trouble piecing together what  if translated to actual sensical statements  you are objecting to:
1.) Existential Assumptions
You are upset about the empty set, calling it a fixed point  though, you do not complain about the axiom of infinity (are you maybe not aware of it?). However, you also object to the axiom of choice; but I can't tell if this is because you think the axiom of choice posits existence is a way other axioms don't, or because of a confusion over the other axioms. For example, the axiom of pairing says that if you have some sets, then a certain other set exists; the axiom of choice does the same thing  so, where you not aware that it operates in the same way and thought that it baldly posited existence, or is there something specific about choice?
2.) Model Theory
You seem to be confusing axioms as logical statements and things like the cumulative hierarchy (you kind of allude to this, I think). Is your objection, perhaps, not to axioms, but to Platonic attitudes towards V? More generally (not exactly relatedly), do you have some general objection to model theoretic perspectives, philosophically speaking? For example, is it alright to define a group axiomatically, and proceed formally, but not to reason by assuming that we are working in some generic group? Essentially, is your complaint not that we can manipulate axioms formally as symbols; but that we talk about things that satisfy those axioms? I realize that this is a bit vague
3.) Existence that you think isn't "constructive"
Is choice a problem because I can't always construct  in some sense  a choice set? Does this somehow apply to the empty set too  or does position 1 mix in with this?
4.) Statements that don't have an empirical sense
This seems an option based on allusion to physics, but I can't figure out exactly what you're trying to say with it.

Is your basic thesis, "Abstract objects don't exist, only physical ones; any approach that assumes symbolic manipulation refers to anything, thus, is bogus. However, we can talk about objects in relation to objects, so long as at no point do we posit that any of those objects, actually, are."
This is what it feels like you want to say; but, if that's true, then you are attacking things the wrong way and wrong headedly. If it is not what you are trying to say, then I've got nothing.
Finally, I know you don't like me  you think I'm a trolling jerk (and, I admit, I've had fun at your expense because I think you kind of deserve it)  but, this post is entirely in earnest  that said, while I would like a response and am willing to discuss; I do not endorse any of these ideas, nor am I giving them credence (nor your attitude) by so doing.
1.) Existential Assumptions
You are upset about the empty set, calling it a fixed point  though, you do not complain about the axiom of infinity (are you maybe not aware of it?). However, you also object to the axiom of choice; but I can't tell if this is because you think the axiom of choice posits existence is a way other axioms don't, or because of a confusion over the other axioms. For example, the axiom of pairing says that if you have some sets, then a certain other set exists; the axiom of choice does the same thing  so, where you not aware that it operates in the same way and thought that it baldly posited existence, or is there something specific about choice?
2.) Model Theory
You seem to be confusing axioms as logical statements and things like the cumulative hierarchy (you kind of allude to this, I think). Is your objection, perhaps, not to axioms, but to Platonic attitudes towards V? More generally (not exactly relatedly), do you have some general objection to model theoretic perspectives, philosophically speaking? For example, is it alright to define a group axiomatically, and proceed formally, but not to reason by assuming that we are working in some generic group? Essentially, is your complaint not that we can manipulate axioms formally as symbols; but that we talk about things that satisfy those axioms? I realize that this is a bit vague
3.) Existence that you think isn't "constructive"
Is choice a problem because I can't always construct  in some sense  a choice set? Does this somehow apply to the empty set too  or does position 1 mix in with this?
4.) Statements that don't have an empirical sense
This seems an option based on allusion to physics, but I can't figure out exactly what you're trying to say with it.

Is your basic thesis, "Abstract objects don't exist, only physical ones; any approach that assumes symbolic manipulation refers to anything, thus, is bogus. However, we can talk about objects in relation to objects, so long as at no point do we posit that any of those objects, actually, are."
This is what it feels like you want to say; but, if that's true, then you are attacking things the wrong way and wrong headedly. If it is not what you are trying to say, then I've got nothing.
Finally, I know you don't like me  you think I'm a trolling jerk (and, I admit, I've had fun at your expense because I think you kind of deserve it)  but, this post is entirely in earnest  that said, while I would like a response and am willing to discuss; I do not endorse any of these ideas, nor am I giving them credence (nor your attitude) by so doing.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
Re: Axiomatic mathematics has no foundation
Schrollini wrote:Treatid wrote:Schrollini wrote:These things are called sets.
I mostly agree. In practice, it has been decided that a set can't contain itself... (at least in some set theories) which is one of those impossible fixed points. And I feel that sets themselves are a bit inefficient. But as a generic object that represents relationships then sets are absolutely already doing what I want. If someone constructs a set theory with just sets, and no fixed points, then I will happily embrace that theory... even if they insist on calling it an axiomatic theory.
Care to point me to the axiom in ZF that says sets can't be elements of themselves? I can't find it.
The axiom of regularity, implies that a set can't be an element of itself.
The axiom states that every nonempty set A contains an element that is disjoint from A.
@Treatid:
You said that by replacing the "empty set" by "some set" makes one unable to construct the number systems, that's false. If that's the only axiom you change, you can still get the empty set.
 Forest Goose
 Posts: 377
 Joined: Sat May 18, 2013 9:27 am UTC
Re: Axiomatic mathematics has no foundation
Demki wrote:Schrollini wrote:Treatid wrote:Schrollini wrote:These things are called sets.
I mostly agree. In practice, it has been decided that a set can't contain itself... (at least in some set theories) which is one of those impossible fixed points. And I feel that sets themselves are a bit inefficient. But as a generic object that represents relationships then sets are absolutely already doing what I want. If someone constructs a set theory with just sets, and no fixed points, then I will happily embrace that theory... even if they insist on calling it an axiomatic theory.
Care to point me to the axiom in ZF that says sets can't be elements of themselves? I can't find it.
The axiom of regularity, implies that a set can't be an element of itself.
The axiom states that every nonempty set A contains an element that is disjoint from A.
@Treatid:
You said that by replacing the "empty set" by "some set" makes one unable to construct the number systems, that's false. If that's the only axiom you change, you can still get the empty set.
The axiom of foundation (regularity) is independent of ZF, and can be replaced with axioms that lead to a nonwellfounded theory* (see: http://standish.stanford.edu/pdf/00000056.pdf). The main problem with sets not containing themselves is Russel's Paradox, which is avoided by the axiom of separation. Although, there are other methods of avoiding the paradox: Quine's New Foundations (I'm not sure about where the consistency of this falls relative other theories  I'm not sure it's established) and Positive Set Theory both avoid Russel's paradox, but have things like universal sets and other such.
*As in "a theory of nonwellfounded sets" not a theory that isn't well founded, obviously:)
**This is not meant to be contentious, but informative to anyone who night be interested since I think it's neat  my tone is off sometimes, I'm not sure which way it reads.
Last edited by Forest Goose on Fri Mar 07, 2014 8:23 pm UTC, edited 1 time in total.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
 Schrollini
 Posts: 515
 Joined: Sat Sep 29, 2012 5:20 pm UTC
Re: Axiomatic mathematics has no foundation
Demki wrote:The axiom of regularity, implies that a set can't be an element of itself.
The axiom states that every nonempty set A contains an element that is disjoint from A.
Thanks for clarifying. The point I was trying to make is that ZF doesn't contain the axiom ∀x(x ∉ x). This is implied by the axiom of regularity and the axiom of pairing, as the empty set is implied by the axiom schema of specification. Treatid seems concerned that ZF contains "arbitrary" rules about existence and nonexistence of some things. I'm trying to argue that it doesn't; it just contains rules about the types of sets that can exist; from those we can derive the those "arbitrary" existence rules.
But I obviously didn't manage to get that point across, so thanks straightening things out.
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Re: Axiomatic mathematics has no foundation
This old flame might be at least interesting: http://golem.ph.utexas.edu/category/201 ... heory.html
TLDR: category theoretic, rather than set theoretic axioms. The axioms are much less contentious: we pretty clearly can't get rid of any of them without breaking what we think of as [classical] maths. It turns out to be equivalent to some slightly weaker version of ZFC (and with an eleventh axiom, equivalent).
In general, I suspect category theory to be a cleaner foundation. We may take issue with some of the nitty gritty behaviors of sets, but we generally agree on the desired behaviors of things like functions.
TLDR: category theoretic, rather than set theoretic axioms. The axioms are much less contentious: we pretty clearly can't get rid of any of them without breaking what we think of as [classical] maths. It turns out to be equivalent to some slightly weaker version of ZFC (and with an eleventh axiom, equivalent).
In general, I suspect category theory to be a cleaner foundation. We may take issue with some of the nitty gritty behaviors of sets, but we generally agree on the desired behaviors of things like functions.
Re: Axiomatic mathematics has no foundation
treatid wrote:The difference lies in understanding what is being done.
But all mathematicians do understand what's being done. Everybody knows that if you keep deconstructing the meaning of what you're doing, it devolves into nonsense. Mathematicians know that mathematics is not immune to this sort of thing. When a mathematician talks about "proof", what they mean is "a proof good enough to convince me". They don't mean that what's being demonstrated could stand up to the scrutiny of philosophy, because nothing can. We're just discussing objects which may or may not exist that we find beautiful, interesting, and useful. We've developed language to discuss and reason about these objects, and if you came out tomorrow with an argument that the foundation of mathematics wasn't consistent, the bulk of mathematicians would just keep on working as usual, while a small minority would turn to giving a philosophically sound basis for the way we express the ideas. Unless, of course, you managed to find the inconsistency using "everyday" mathematical objects. That would cause a bigger stir, I imagine.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.
 Forest Goose
 Posts: 377
 Joined: Sat May 18, 2013 9:27 am UTC
Re: Axiomatic mathematics has no foundation
z4lis wrote:treatid wrote:The difference lies in understanding what is being done.
But all mathematicians do understand what's being done. Everybody knows that if you keep deconstructing the meaning of what you're doing, it devolves into nonsense. Mathematicians know that mathematics is not immune to this sort of thing. When a mathematician talks about "proof", what they mean is "a proof good enough to convince me". They don't mean that what's being demonstrated could stand up to the scrutiny of philosophy, because nothing can. We're just discussing objects which may or may not exist that we find beautiful, interesting, and useful. We've developed language to discuss and reason about these objects, and if you came out tomorrow with an argument that the foundation of mathematics wasn't consistent, the bulk of mathematicians would just keep on working as usual, while a small minority would turn to giving a philosophically sound basis for the way we express the ideas. Unless, of course, you managed to find the inconsistency using "everyday" mathematical objects. That would cause a bigger stir, I imagine.
I would take this further (personally, speaking):
I would say that a proof warrants accepting that a theorem is mathematically true, and that if it is, actually, mathematically true hinges on if there exists a consistent system in which the proof can be carried out in. I would say that it is the role of the philosopher to examine what exactly mathematically true translates to in a broader sense; but, whatever they ultimately decide, I don't think any of that modifies the character of the result as an object of mathematics, only in terms of the broader relation of mathematics to things that are not mathematics.
An analogy:
If I say that "Samantha has pretty eyes", this expresses something true in terms of reality as it exists at the level of experience accessible to me. If someone comes up, later, with an airtight argument that we are in the matrix, or brains in a vat, or some such, this doesn't negate/modify anything about my statement at the level it was expressed  on a larger level, though, what exactly that statement may express may be different than what I meant. But, I wouldn't accept the argument that since "Samantha" is actually a computer program in the matrix and doesn't have eyes that what I said suddenly becomes nonsense/false, it would just mean that what I meant by terms like "Samantha", "eyes", "pretty", etc. refer in a different way than I originally realized. I don't believe that you can do a philosophical analysis irrespective some level of discourse, in the sense discussed, as concerns the truth/falsity of an individual expression  perhaps, it is brains in vats all the way down (up?), forever and ever, that shouldn't entail that I can't make statements relative some level of the system*, and I don't think that it does.
In the same sense, grand philosophical arguments may be held over what exactly we are referring to (,or not referring to), when we say "1 + 1 = 2"**, but the ultimate outcome of those arguments does not suddenly change the character of the statement inside the original system (as a statement of mathematics); and I don't believe that mathematicians state theorems, or ought consider theorems (as a mathematical object), outside of their capacity as mathematical objects  anymore so than I ought/do consider the prettiness of my girlfriend's eyes outside the scope of my experience and at the level that she may be a computer simulation. I feel like responding to mathematics by undercutting it philosophically is like arguing with someone who asserts they love their spouse by pointing out that philosophical skepticism has never been perfectly refuted  it's not that you'd be wrong to say it, it's that you'd be missing the content and the context; much like when someone takes a joke/sarcasm as serious, they aren't wrong, but they surely are missing the point of the whole activity.***

*Even that very sentiment still suffers the same constraints, it holds up at the level of the whole system of brains in vats as a single system, but maybe, again, that system is embedded in some greater context. There is always going to be a level of discourse, trying to talk fully absolutely is like arguing that logic fails  it doesn't provide insight, nor does it enlarge understanding/perspective, it simply entails that you must shut up and say nothing. (see ***)
**Even in mathematics (as in everything) there is a stratification of levels of discourse and what it refers to. For example, "1 + 1 = 2" is not accurate if I am talking in the larger context of groups, the integers (mod 2)  in the strictest sense  do not have such a theorem. A better example might be "1 + 2 = 2 + 1", which no longer holds once the context is enlarged to include nonabelian objects.
***I feel that we unfairly reduce a lot of pursuits down to the wrong concepts. The pursuit of mathematics is not "Find absolute truth", anymore than the pursuit of art is "Find absolute beauty", it is to expand upon and increase understanding and perspective and to generate greater insight into the nature of the thing  art gives us a keener insight into the artistic nature of its object, mathematics gives us a keener insight into the mathematical nature of its object; and criticizing mathematics on the level of a philosophically external scrutiny over the absolute nature of its truth is a lot like scrutinizing a poem over the literal accuracy of its phrases. Finally, as to what exactly "artistic nature" or "mathematical nature" mean, that too is an activity for philosophers; we (outside philosophy) already know it when we see it, even if we can't immediately define it (and we don't need to)  much in the same sense as I know when a video I'm viewing is pornography, sans a perfect definition; and, to be crass, I do not need to study the philosophy of porn for a few years to feel aroused by it  the same, albeit less vulgarly, applies here.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
Re: Axiomatic mathematics has no foundation
Treatid, AFAIS, you're upset about that we don't have objective knowledge, and that the sciences aren't absolute, which is to say, has always been.
Re: Axiomatic mathematics has no foundation
Forest Goose wrote:An analogy:
If I say that "Samantha has pretty eyes", this expresses something true in terms of reality as it exists at the level of experience accessible to me. If someone comes up, later, with an airtight argument that we are in the matrix, or brains in a vat, or some such, this doesn't negate/modify anything about my statement at the level it was expressed  on a larger level, though, what exactly that statement may express may be different than what I meant. But, I wouldn't accept the argument that since "Samantha" is actually a computer program in the matrix and doesn't have eyes that what I said suddenly becomes nonsense/false, it would just mean that what I meant by terms like "Samantha", "eyes", "pretty", etc. refer in a different way than I originally realized. I don't believe that you can do a philosophical analysis irrespective some level of discourse, in the sense discussed, as concerns the truth/falsity of an individual expression  perhaps, it is brains in vats all the way down (up?), forever and ever, that shouldn't entail that I can't make statements relative some level of the system*, and I don't think that it does.
In the same sense, grand philosophical arguments may be held over what exactly we are referring to (,or not referring to), when we say "1 + 1 = 2"**, but the ultimate outcome of those arguments does not suddenly change the character of the statement inside the original system (as a statement of mathematics); and I don't believe that mathematicians state theorems, or ought consider theorems (as a mathematical object), outside of their capacity as mathematical objects  anymore so than I ought/do consider the prettiness of my girlfriend's eyes outside the scope of my experience and at the level that she may be a computer simulation. I feel like responding to mathematics by undercutting it philosophically is like arguing with someone who asserts they love their spouse by pointing out that philosophical skepticism has never been perfectly refuted  it's not that you'd be wrong to say it, it's that you'd be missing the content and the context; much like when someone takes a joke/sarcasm as serious, they aren't wrong, but they surely are missing the point of the whole activity.***
This is a good summation of the situation.
There is no objective truth, and if we take our tools to their logical conclusion we find they reveal nothing. Yet our perceptions do exist and are real for us.
There is a positive spin to this. First and foremost: While there is no absolute truth... there is meaning with respect to an observer. We need to make ourselves (the observer) an explicit component of the systems being observed. The observer is not a fixed point  but we are a consistent reference. The only way we can understand anything is with respect to ourselves.
Secondly, we know that a thing that cannot be understood by analysing it's component parts is an emergent property. We have (relatively) recently learned about complex systems and emergent properties. We can now see that everything we perceive is an emergent property. An axiomatic approach to understanding is simply the wrong approach to knowledge. Not only can we not have axioms, but neither can we construct knowledge in an 'A therefore B' fashion. To cling to the axiomatic structure as a source of knowledge is... silly. But we already have a rough knowledge of how emergence comes about. We know that the mechanisms of emergence are understandable.
Twistar has a thread on Circularity in Formal Languages. There it is agreed that reducing the required assumptions to a minimum in some way legitimises the necessary reliance on informal languages to state those assumptions.
"We only need a couple of assumptions to build all of mathematics".
Tough. You can't have a couple of assumptions. No axioms means no axioms.
Mathematics can describe patterns of relationships. And those patterns sometimes have a significance for us humans. The art of mathematics is to find patterns of relationships that we find meaningful. In this, mathematics is fully justified. It works. What isn't justified is to claim that there is a definite logic involved in this process or that any perceived results are a direct consequence of the starting state.
lalop wrote:This old flame might be at least interesting: http://golem.ph.utexas.edu/category/201 ... heory.html
TLDR: category theoretic, rather than set theoretic axioms. The axioms are much less contentious: we pretty clearly can't get rid of any of them without breaking what we think of as [classical] maths. It turns out to be equivalent to some slightly weaker version of ZFC (and with an eleventh axiom, equivalent).
In general, I suspect category theory to be a cleaner foundation. We may take issue with some of the nitty gritty behaviors of sets, but we generally agree on the desired behaviors of things like functions.
Good call. (I apologise to people who have mentioned Category Theory in my presence previously. I really should have looked into it sooner).
Category Theory is the right sort of foundation... and similar to what I was attempting to describe by reference to digraphs (much more so than ZFC). It is clear to me that Category Theory is a direct recognition of the impossibility of defining axioms.
Schrollini wrote:Treatid seems concerned that ZF contains "arbitrary" rules about existence and nonexistence of some things. I'm trying to argue that it doesn't; it just contains rules about the types of sets that can exist; from those we can derive the those "arbitrary" existence rules.
I feel that there is a boundary that keeps shifting in favour of whatever argument is being made at the time.
There are patterns and sequences of patterns that mathematics can describe. By themselves, these patterns have no meaning and no justification. They are just patterns. Category Theory is well structured to describe those patterns efficiently. ZFC can describe those patterns a little less efficiently. Existing 'axiomatic' systems are just patterns and can be described by either ZFC or Category Theory (or any other Universal Turing Machine Equivalent).
These patterns, being just patterns, don't describe anything. There is nothing inherent in these patterns.
By drawing the boundary around mathematics such that mathematics is only statements of these patterns, you are able to declare that mathematics isn't secretly trying to declare axioms and isn't building on nonexistent properties. I understand this argument and agree that by itself it is a legitimate, sound argument.
The trouble is that this really does leave mathematics as being completely meaningless. All the constructs that are commonly thought of as mathematics: Proof, Euclidean Geometry, Number Systems, Equality,... everything... lies outside mathematics. In practice I see mathematicians drawing the boundary more broadly. Not least, your drawing of the boundary places the axioms of ZF, ZFC and/or Category Theory outside of mathematics.
Nevertheless, I'll accept the boundary as described. It doesn't really change anything. The overall issues still remain. Nothing has been fixed by defining this boundary except to make it "somebody else's problem". It is still impossible to define a set of axioms. It is still impossible to prove anything.
It seems to me to be a somewhat Pyrrhic victory to protect mathematic's integrity by excluding everything useful from mathematics.
PM 2Ring wrote:Axiom  A Little Ray Of Sunshine (1970)
madaco wrote:I can't tell, are you rejecting the idea of talking about a thing, and then giving that thing a name?
Between you and Twistar I needed to really think about this... and the empty set is more of a symptom than the disease.
I came up against a couple of problems in trying to pin down what I was trying to get at.
The first is that definitions can be shifted around pretty arbitrarily. ZFC can be constructed in other languages (systems). Likewise, ZFC can construct most other systems. If I pick on a point as being the source of my qualms, it is generally possible to show that point can be constructed from some other system. I then presumably have to demonstrate the argument for that other system which in turn can be constructed... and so it goes.
If one assumes a fixed point (no matter how well disguised that assumption), then one can construct other fixed points from that source. If one has a loop of points A > B > C > A the accusation of being a fixed point can always be passed off. The result is that examining any one point seems to show that point isn't itself a fixed point... yet the group of points do contain the (hidden) assumption of a fixed point (definition).
It is hard to show that a fixed point is being assumed when the assumption is distributed over a number of related objects.
The other point I've already touched on above. Mathematics can turtle and draw a boundary around the statement of patterns of relationships. This explicitly excuses mathematics of any responsibility for perceived meaning. A valid tactic in itself... but it excludes the statement of axioms from mathematics. It seems to me that (some) people are trying to have it both ways: they argue that the axioms of set theory (or whatever) don't contain impossible assumptions, but exclude those axioms from mathematics leaving no basis on which to judge the claims of lack of impossibility.
And, of course, we know that axioms cannot be defined in the first place.
i) Relationships exist. Whether we call this an axiom or a simple observation, it is what we have.
ii) Everything else is undefinable. Where a set of 'axioms' goes beyond i) that is evidence that impossible assumptions are being made (however, see next).
iii) Not all axioms are definitions per se. We have "the empty set" rather than many empty sets because that choice remains open. As you note, where elements are indistinguishable we are free to treat them as the same object. This doesn't make them the same object... it just means that without distinction there is no harm in treating them as the same object... and doing so tends to make life easier.
iv) If you are constructing something  you have assumed a fixed point. Axiomatic construction only works if it has something to construct from.
v) Witness for prosecution: axiom of regularity. This axiom is preferred over the axiom schema of separation because without the axiom of regularity set theories tend to be too weak. Well d'uh. Without axioms, axiomatic theory can't construct anything. The fact that 'something' is making axiomatic theory appear to work (to any degree) is evidence that 'something' is inextricably connected to an (impossible) fixed point assumption.
vi) Having an explicit empty set contained within every other set but specifically excluding identity relationships (and other loop structures) is bizarre. As much as I do understand the reasoning... It is pretty darned handy to do anything useful with sets... to dispute the existence of identity is a level of denial that I have trouble wrapping my head around.
vii) The axiom of choice is as close to defining 'impossible' as I can imagine. I know that the axiom of choice is controversial and wasn't just accepted without resistance. I also know that it is useful (to define other impossible systems). However, it is now commonly accepted... if mathematicians are that brazen about basing mathematics on flat out impossibilities then I have little faith that they are making sincere efforts to avoid other impossible assumptions.
You might say/argue/state that giving a thing a name suggests "uniqueness" or something, and that that would be misleading or something,
but in pretty much all cases (I have not though of any counterexamples)
calling two things that have the same properties/relations , and as such cannot be distinguished, "the same thing" doesn't have any bad results.
As noted, I'm okay with labelling... although I did spend some time being sure. And treating indistinguishable objects as if they are the same object I'm strongly in favour of  although, again, it is a position I had to justify to myself.
Which is to say  your arguments are very much in the realm of things I'm thinking about.
Although if you want to call it something other than "empty" because you believe that the word has some thing that changes meaning or something, you could do that without issue, other than that it would hinder communication.
The name of the label was bothering me. I was concerned that the empty set was anchoring the construction of V (the von Neumann hierarchy). Having thought about it... I'm currently ambivalent about V. As a construction of all possible unique tree structures I can see a direct correlation with all unique digraphs or all unique bit strings. It isn't obvious to me that I can reject V or the empty set when I accept the possibility of enumerating unique digraphs starting from a single vertex.
However, the exclusion of identity (and subsequent loops) from the structure of V has me concerned. As much as excluding identity seems to be a small detail... the consequences of not excluding identity are significant. It utterly changes the nature (and construction) of V. Vwithidentity is not a simple superset of Vwithoutidentity. Many states that can be unambiguously distinguished within V would no longer be distinguishable.
oh also, do you have any thoughts about tautologies, and general implication stuff?
like do you accept a>a ?
and a V !a
etc.
Tautologies: Without a fixed starting point, all we are left with are tautologies. Given some point (indivisible object) there is nothing we can say about that object (except, maybe, perhaps, possibly, it exists). As such, the only sentence we can construct with respect to the object is necessarily a tautology (or an equally meaningless noncircular statement). However, not all tautologies are the same tautology.
Every instance where we think we are defining a point, we are actually creating a tautology. We are defining that point by reference to other points that are defined by reference to other points that...
However, given both an internal observer and differing patterns in the way tautologies are formed (and a bit of emergence) we can perceive meaning and significance.
More specifically, I see all of existing 'axiomatic' mathematics as being tautological. mathematics hasn't discovered or invented anything. It has just found new and exciting ways of expressing tautologies. This isn't to devalue mathematics... without having a fixed point, there just isn't any other alternative.
So, yes, 'a' does imply 'a'. However, that is all that is directly implied. 'a' says nothing about anything that isn't 'a'.
'a V !a' is more complex. True and False aren't definable. Knowing that True and False are exclusive and complementary doesn't actually tell us anything beyond us having asserted that they cannot be the same object. Nor does '!a' tell us anything if we cannot know what 'a' is. 'a' might be that positron traveling backwards in time and being every electron while doing so. Asserting that '!a' and 'a' must be distinct doesn't actually reveal any information when we cannot know either 'a' or '!a'.
'a or !a' is true. We just don't know what true is... so we haven't actually said anything. But we already knew that there was no objective knowledge.
We can understand things. I'm not trying to be a nihilist. The axiomatic approach to knowledge is fundamentally and fatally flawed. There is no 'A therefore B'. We will gain nothing by trying to construct chains of logic when we cannot possibly specify the beginning of the chain.
But we can understand the relationship between an observer and their context. We can understand how emergent properties arise. We can understand how meaning and significance arise for an observer in a system that supports emergence.
In the meantime, mathematicians don't need to understand what they are doing. With trial and error they will occasionally stumble upon something we feel is of relevance. Their belief in axiomatic approaches is a slight handicap.
@Twistar: Your argument is persuasive. I'm now of the opinion that the empty set cannot be empty. I think the identity relationship is too fundamental to exclude. However, you are correct that objects with just identity relationship are specifiable. The empty set is not the bad guy I was claiming.
doogly wrote:The fifth postulate was extremely explicit. That is why it is called the fifth postulate; in Euclid's Elements, it is the fifth one listed. It was right there in the text.
But it wasn't known that it definitely needed to be a separate axiom until relatively recently. Else nonEuclidean Geometry would have been a thing since the inception of Euclidean Geometry.
And let's focus then. You are saying that the statement, "An empty set exists," is problematic. It can't be used in any foundational way. Why is that? Do you think there is a communication barrier? If Schrollini says "empty set," that without some angel to whisper truths in my ear, I cannot figure out what the "meaning" of empty set is?
Actually... yes, sort of. Transdimensionalalien time: without any common ground we have no way to start a conversation. You and Schrollini share enough common experiences that you can often come to an apparent agreement. However, that apparent agreement does not imply that either of you understand the concept in an absolute sense  you can't.
For your purposes as humans on Earth, rough agreement over "meaning" is good enough most of the time. But if we push the boundaries, if we examine edge cases, rough agreement, even well refined but not complete agreement, is insufficient.
Looking at Quantum Mechanics... we have pretty good evidence that the microworld is not just a miniature version of our human scale experience on the surface of a planet. The networks of assumptions and meanings built through interactions as a human don't hold for photons and electrons.
'course... I was actually wrong in picking on the empty set.
sonicspin wrote:Treatid, AFAIS, you're upset about that we don't have objective knowledge, and that the sciences aren't absolute, which is to say, has always been.
I agree that this is the way it has always been. And I'm fine with that. I think it is interesting that we have the limit on what and how we can know things.
I think that we should understand this limitation and adapt our pursuit of knowledge accordingly. Axiomatic mathematics is a dead end. Axioms and the associated proofs and logic don't exist. So let us construct systems that are aligned with the known limitations.
Re: Axiomatic mathematics has no foundation
Treatid wrote:Axiomatic mathematics is a dead end. Axioms and the associated proofs and logic don't exist.
...And yet you write this statement using a computer created by people utilizing the logic you claim does not exist.
Humans can make themselves immortal and become gods of the universe  achieve anything and everything that is achievable  all without necessity to avoid axioms.
If that's a 'dead end' bring it on...
 doogly
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Re: Axiomatic mathematics has no foundation
No, they knew you needed to explicitly include it as a postulate. That is why it was a numbered postulate, the numbering was the making explicit.
What was less clear was that doing without it was perfectly fine and delightful. They knew you needed it to get Euclidean geometry, they just considered the things you got without it unacceptable.
What was less clear was that doing without it was perfectly fine and delightful. They knew you needed it to get Euclidean geometry, they just considered the things you got without it unacceptable.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 Forest Goose
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Re: Axiomatic mathematics has no foundation
This is a good summation of the situation.
It is, I think; but I also think that it is summing something different than what you happen to think. A lot of your argument is analogous to when someone just keeps responding with "why". They are not being clever, they are being the opposite of it.
There is no objective truth, and if we take our tools to their logical conclusion we find they reveal nothing.
This is absolute rubbish  if your definition of truth and knowledge lead you to the conclusion that there are no such things, then it is not that you've found something deep, it's that you've used the wrong definitions. My ass is currently in my chair, this is objectively true  "my ass" and "my chair" may have a very different fundamental nature than I expect, but it doesn't change anything  indeed, long before anyone knew about the particle physics of asses and chairs, people were still correct in asserting just where their asses were. In short, an infinite tower of "why"'s does not negate anything.
The main point: philosophy answers to sanity, not madness  epistemology is about uncovering the nature of knowledge, and you can definitely make up neat theories that seem to throw everything into doubt, but that's more of a Barber Paradox situation than anything useful: in that it's not really paradoxical, but a good reason to expect that you've messed up in your premises.
.There is a positive spin to this. First and foremost: While there is no absolute truth... there is meaning with respect to an observer. We need to make ourselves (the observer) an explicit component of the systems being observed. The observer is not a fixed point  but we are a consistent reference. The only way we can understand anything is with respect to ourselves.
There is a subtle equivocation here  that "1 + 1 = 2" is a mathematical truth means that at the level of those things, whatever they may be, that that holds true  "1 + 1 = 2" is true for formalists and for platonists, they differ in what "1" and "2" are, but that's not really relevant. You're talking about a very different kind of relativity, it sounds quite subjective and I don't see any basis for it. We don't have absolute knowledge is true in so far as concepts always, at some point, must be bracketed off and treated functionally (, which, ends up meaning axiomatically and formally); this does not entail some weird dependence on the subject who comes across such things.
Secondly, we know that a thing that cannot be understood by analysing it's component parts is an emergent property. We have (relatively) recently learned about complex systems and emergent properties. We can now see that everything we perceive is an emergent property. An axiomatic approach to understanding is simply the wrong approach to knowledge. Not only can we not have axioms, but neither can we construct knowledge in an 'A therefore B' fashion. To cling to the axiomatic structure as a source of knowledge is... silly. But we already have a rough knowledge of how emergence comes about. We know that the mechanisms of emergence are understandable.
Since I have yet to see any type of convincing argument for any of this, or even hard definitions, it all sounds like fluff. You speak like a politician.
Tough. You can't have a couple of assumptions. No axioms means no axioms.
This would make sense...if you had compelling arguments, were an authority, and this was a widely accepted position; but none of that is the case. I don't understand why you suddenly get to make up the rules.
tl;dr
I still strongly disagree, I still don't think you've presented any arguments or definitions, and I still don't think you understand what anyone else here is saying.
Last edited by Forest Goose on Sun Mar 16, 2014 4:12 am UTC, edited 1 time in total.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

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Re: Axiomatic mathematics has no foundation
@Treatid: I think you fundamentally misunderstand the purpose of mathematical axioms. They're not intended to define some sort of groundlevel absolute knowledge from which all else springs forth. They're not intended as assertions that you have to assume are "true" in some platonic sense or that you have to take on faith. And there's no problem if they can't be "defined" in the way that you keep using the word "define". I can only guess that you think that mathematicians are trying to use axioms for something roughly along these lines, and that you've been implicitly setting up and attacking a straw man this whole time by arguing that this doesn't work, when axioms never served this purpose in the first place.
The axioms of any mathematical system are like the rules of Chess. All they do is describe the rules of a hypothetical system that we might want to study and play with at a given time, the same way that the rules of Chess describe a particular game that we might want to study and play at a given time. Theorems are simply things that people have deduced about how a particular system must behave as a logical consequence of these rules, the same way that the rules of Chess imply that certain positions are guaranteed to be won or lost if players play correctly. Math is nothing more than the study all the different varieties of these systems whose rules we've made up. And often we can apply the knowledge gained to the real world  whenever some aspect of reality behaves in a way that closely matches the rules of a mathematical system we've studied, we can then bring to bear to all the different mathematical tools and theorems we've developed for that system to help solve problems involving that aspect of reality. Not surprisingly, some of the areas in math most heavilystudied are the ones that are useful for modeling and solving problems in the real world.
Arguing that all the math currently studied so far is foundationless and suspect because the axioms can't be properly "defined" is akin to arguing to a Chess player that all of his knowledge and experience in the game is foundationless and suspect because the rules of Chess can't be properly "defined". It's obviously a silly thing to say. Not having some universal absolute reference point for communicating the rules of Chess has not stopped the development of lots of interesting theory and knowledge about Chess, and it's laughable to claim that the task of *defining and agreeing what the rules are* is the major obstacle involved. The same goes for math.
The axioms of any mathematical system are like the rules of Chess. All they do is describe the rules of a hypothetical system that we might want to study and play with at a given time, the same way that the rules of Chess describe a particular game that we might want to study and play at a given time. Theorems are simply things that people have deduced about how a particular system must behave as a logical consequence of these rules, the same way that the rules of Chess imply that certain positions are guaranteed to be won or lost if players play correctly. Math is nothing more than the study all the different varieties of these systems whose rules we've made up. And often we can apply the knowledge gained to the real world  whenever some aspect of reality behaves in a way that closely matches the rules of a mathematical system we've studied, we can then bring to bear to all the different mathematical tools and theorems we've developed for that system to help solve problems involving that aspect of reality. Not surprisingly, some of the areas in math most heavilystudied are the ones that are useful for modeling and solving problems in the real world.
Arguing that all the math currently studied so far is foundationless and suspect because the axioms can't be properly "defined" is akin to arguing to a Chess player that all of his knowledge and experience in the game is foundationless and suspect because the rules of Chess can't be properly "defined". It's obviously a silly thing to say. Not having some universal absolute reference point for communicating the rules of Chess has not stopped the development of lots of interesting theory and knowledge about Chess, and it's laughable to claim that the task of *defining and agreeing what the rules are* is the major obstacle involved. The same goes for math.

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Re: Axiomatic mathematics has no foundation
All right, I'm diving in.
Treatid, you say that "without fixed points, all we are left with is tautologies". You've also acknowledged that it's possible to define rules of symbol manipulation, although those symbols don't mean anything.
The point people have been trying to get across to you is that that's all mathematics tries to do in the first place.
Axioms aren't "fixed points". Axioms are just the stuff we put on the left side of a tautology.
You've acknowledged that something like "A>A" can be recognized as a tautology. When a mathematician claims to have "proven" some theorem in ZFC, that's just a linguistic shorthand for saying "(axioms of ZFC)>(theorem of ZFC) is a tautology". Studying ZFC isn't a matter of assuming the axioms have some real meaning or that they are "true" in some sense. Studying ZFC just means asking the question "what kind of tautologies can I get that are of the form (axioms of ZFC)>(something)?"
You yourself have acknolwedged that tautologies are allowed, and that not all tautologies are the same as each other. So there should be nothing wrong with wondering about tautologies that follow some particular format. That also fits with your position that we should only focus on relationships between things without any notion of what the things themselves are. Just as "A>A" is a tautology no matter what A might happen to be, even if A is completely meaningless, likewise "(axioms about sets)>(theorem about sets)" is a tautology no matter what a "set" might happen to be, even if "set" is a meaningless term.
That's all mathematics does. The *application* of mathematics is where we ascribe meaning. Somebody says to himself "if I interpret these arbitrary symbols of 'calculus' as a description of how the electromagnetic force works, then this other arrangement of the same symbols, as permitted by 'calculus', would be a description of what happens in this particular physics problem". The connection between math and our mental intuitions of quantifying things is something we do as an application of math. That's not what math is in and of itself.
When mathematicians talk about things being "proven" or about sets "existing" or about things being true given a set of axioms, all that is is a linguistic shorthand for describing which tautologies we find interesting. You may have a point in complaining about overly Platonic attitudes among mathematicians, but that would be a social problem, not a problem in how mathematics itself is structured.
Treatid, you say that "without fixed points, all we are left with is tautologies". You've also acknowledged that it's possible to define rules of symbol manipulation, although those symbols don't mean anything.
The point people have been trying to get across to you is that that's all mathematics tries to do in the first place.
Axioms aren't "fixed points". Axioms are just the stuff we put on the left side of a tautology.
You've acknowledged that something like "A>A" can be recognized as a tautology. When a mathematician claims to have "proven" some theorem in ZFC, that's just a linguistic shorthand for saying "(axioms of ZFC)>(theorem of ZFC) is a tautology". Studying ZFC isn't a matter of assuming the axioms have some real meaning or that they are "true" in some sense. Studying ZFC just means asking the question "what kind of tautologies can I get that are of the form (axioms of ZFC)>(something)?"
You yourself have acknolwedged that tautologies are allowed, and that not all tautologies are the same as each other. So there should be nothing wrong with wondering about tautologies that follow some particular format. That also fits with your position that we should only focus on relationships between things without any notion of what the things themselves are. Just as "A>A" is a tautology no matter what A might happen to be, even if A is completely meaningless, likewise "(axioms about sets)>(theorem about sets)" is a tautology no matter what a "set" might happen to be, even if "set" is a meaningless term.
That's all mathematics does. The *application* of mathematics is where we ascribe meaning. Somebody says to himself "if I interpret these arbitrary symbols of 'calculus' as a description of how the electromagnetic force works, then this other arrangement of the same symbols, as permitted by 'calculus', would be a description of what happens in this particular physics problem". The connection between math and our mental intuitions of quantifying things is something we do as an application of math. That's not what math is in and of itself.
When mathematicians talk about things being "proven" or about sets "existing" or about things being true given a set of axioms, all that is is a linguistic shorthand for describing which tautologies we find interesting. You may have a point in complaining about overly Platonic attitudes among mathematicians, but that would be a social problem, not a problem in how mathematics itself is structured.
Re: Axiomatic mathematics has no foundation
Technically, axiomatic theory does not exist. Neither axioms nor any logic or proof following on from axioms can be stated.
This isn't me making a bizarre argument. This is well understood mathematics. Albeit not well understood by everyone.
Obviously mathematicians are doing something that frequently gets labelled as axiomatic mathematics. This is fine, so long as everyone remembers that what is labelled 'axiomatic mathematics' has absolutely nothing to do with the technical definition of axioms and axiomatic mathematics.
Having said that... it can be really difficult to see past our assumptions. To whit: I made a mistake.
Of course the empty set is an attempt to specify a fixed point.
If you assume counting exists then, bizarrely enough, counting exists. Zero or empty are not supposed to be arbitrary labels. They are supposed to be properties that can be uniquely recognised.
Being able to count the number of relationships is proof positive that you are assuming a fixed point.
Set theorists should know this.
As tempting as it is to think that there is a definite nothing against which something can be counted... just thinking zero means that you have already accepted 1 and 2.
Assume that all objects have an implied self reference and that there is no fundamental way to distinguish between two objects or between two relationships  the same logic that gives us THE empty set rather than AN empty set.
We have a ring of 5 objects A>B>C>D>E>A.
Each object in that ring is linked to the next item in the ring and itself. Start at a point on the ring. Look backwards along the ring. Look forward along the ring.
What would be the difference in your observation if the ring was now six elements long?
Seven? A hundred? A million? Just one single element in the ring?
Without any external reference, there is no way to distinguish between rings of different sizes.
Why do you think set theory fights so hard to prevent identity relationships?
If you assume that you can count the number of relationships that an object has (the number of elements in a set), then you don't need ZFC to justify number systems to you. You've already assumed the existence of counting and an ability to distinguish one number from another number.
The whole point of not being able to define axioms is that you can't define axioms. Being able to identify a particular set by any property whatsoever is presupposing some form of identifier.
Any start point can only prove itself. ZFC gives you number systems because you assume that you can count an empty set.
Set theory knows this. If you permit identity relationships and don't start with the counting system you are supposed to be proving... then... well... you can prove diddly squat.
The contortions that set theory goes through to pretend it isn't making any assumptions would be endearing if it didn't involve so many grownass men.
To be fair... I was buying into the delusion myself. I mean, “empty” is so obvious... so self evident...?
And I absolutely can count up to ten on my fingers and toes.
But axiomatic theory isn't common sense. Axiomatic theory is built upon the idea that knowledge can be manipulated in a deterministic fashion. It is a compelling narrative. It seemed plausible. But in practice it relies on you already assuming whatever it is you are trying to show.
For all the people who seem to think that they already know and understand this... what are those outstanding millennial prizes about?
Mathematics already has the answer. Axiomatic theories cannot be stated or defined in any way.
No ifs, buts or maybes...
What is left is just a matter of consensus. 1+1=2 because, by and large, give or take we agree that it should. There is no logic. There is only what works. And sometimes, things only work because everybody is agreeing that it is working and it gets completed before it fails.
And Uncle Sam is quite definite that any... leeway... in thinking will be in his favour.
I do credit axiomatic theory with being intelligent enough to be able to demonstrate that it was completely the wrong way to approach knowledge.
In the mean time, somebody should really sweep up that four million dollars that they are giving away because they apparently didn't notice what was obvious to all other mathematicians... there is no axiomatic mathematics. At all.
As much as people are justifying it because it works... so did sending children down the mine and up chimneys.
Ahem. Before we get too deep into the “bad”, “old”, “obsolete” maths versus the “shiny”, “good”, “healthier” mathematics... (too late?)
...
We have already learned from that old, fangled, math, malarkey that we can distinguish some things... but if we can't definitely distinguish two things then we are free to consider them the same thing. (Hence THE empty set).
Using set theory with an explicit self relationship for every set and no empty set we find that vast numbers of systems cannot be distinguished. It isn't just simple rings that can't be distinguished from a single point... all symmetric structures are indistinguishable within set theory.
A complete digraph of twenty elements is indistinguishable from a ring of a billion elements (again – within set theory and with no external reference).
We may choose to minimise redundancy and regard all such possible structures as one representative... which tells us nothing about whether there really is just a self reference or actually umpteen bazzillion different elements. We just can't do anything to distinguish between the possibilities so we assume a particular possibility to taste...
This is about what we can communicate directly. It doesn't matter what we think we know... it matters what we can communicate. Without a fixed point, symmetry provides no distinguishing features. Counting the number of relationships requires that we can uniquely distinguish relationships.
So... we need a distinguishing feature. We need an asymmetry.
Imagine one of the absurdly big infinities that set theory invented to make itself feel good... imagine that many points all related to each other. And imagine a single self referential point.
We have no way of distinguishing between these two things (in set theory).
Once we remove the artificial fixed points, set theory cannot distinguish between any symmetric systems. All symmetric systems are indistinguishable from a single set which contains itself (identity).
So, obviously the good stuff is where it isn't symmetric.
As an aspiring god looking to create “not symmetric” stuff it can be a bit difficult working out how to begin.
Our first attempts saw us trying to construct things in a logical fashion... but we couldn't establish a fixed point and without a fixed starting point our actions were of no significance.
Symmetry gets boring real fast since, for all practical purposes, it is no different than identity.
So... lets create us some of the assymmetry... let's see... we just grab this bit and... oh yeah... we can't distinguish that bit from any other bit.... hmmm...
Well... I'll invent a difference... I'll say that A is not B. There!
But B is not A. I can't distinguish between A and B. They appear identical in every respect even though I asserted that they were opposites.
Okay, so we need a third point. Each point balances the other two and isn't directly equal the other.... hang on... three points of an equilateral triangle are indistinguishable...
Dang. Assymetry is more difficult than it looks.
Bugger this for a game of billiards... cheat.
Relationships are asymmetric.
Asymmetry is necessary and exists. Even the saltiest dog needs an occasional bone. The existence of relationships and their directionality are the irreducible components of knowledge. Whether or not justified as axioms... all of knowledge must be representable with these components. All languages function by describing relationships.
There is an argument to be made that the existence of asymmetric relationships is a fixed point upon which we can build. An argument I have considerable sympathy for. Even if we can't describe relationships or asymmetry except by reference to themselves... they still present a (necessary) starting point.
Big shout out to Category Theory for putting the functors up front and recognising that objects are just placeholders between the real action.
We've already seen that all symmetric systems are indistinguishable. We can take this one step further... without a method to uniquely distinguish relationships, a given object cannot be identified by counting the number of relationships. Which leaves us no way to distinguish between sets.
Let us jeer at silly set theories that believed identifying sets was of any relevance to anything (at all).
Yay for Category Theory that focuses on the only stuff we can actually specify... the shape of relationships.
One placeholder is completely indistinguishable from another placeholder. By the same token, one relationship is indistinguishable from another relationship.
Axiomatic logic has taught us to try to understand the placeholders and individual relationships. There is nothing to be gained by trying to understand individual objects or relationships. Their internal structure is permanently outside our ability to perceive or communicate.
So... we know that all of everything is a pattern of relationships.
No... seriously... all of everything is a pattern of relationships.
Physics is a pattern of relationships.
There are no fundamental objects. Trying to define photons, electrons, or any other supposed elementary particle, is to not understand mathematics.
The particle centric view of the universe blinds physicists to properly understanding their subject.
Quantum Mechanics supposes that there is an absolute reference frame... apart from assuming a fixed reference frame... this hides the question of what the stuff between particles is.
Defining particles with respect to dimensions is only useful if you have a complete understanding of dimensions.
And we know that we don't have a complete understanding of dimensions... because that would require that we could define them axiomatically. Really... this no axioms stuff has real world implications.
As successful as Quantum Mechanics is at pattern matching... it tells us nothing about the fundamental nature of the universe.
Space is not just some background against which particles do the fun stuff.
The process that results in what we observe as particular particles interacting is what defines space just as much as it defines the particles.
Explaining things in terms of 'spin', 'charge', 'mass' doesn't explain anything.
As a job of pattern matching... Quantum Mechanics is impressive. As a tool for understanding the universe it is a shit hot mess (that isn't a compliment).
Fundamental physics is as far from common human experience as it is possible to get. It is simply not sane to try to describe fundamental processes in terms of concepts that we know cannot be defined.
At our level of technology our theory of physics must have sufficient fidelity to be credible. We are good enough at pattern matching to find a pattern with a good degree of coincidence with observed reality. As an exercise in mathematics (specifically pattern matching) there is nothing remarkable about Quantum Mechanics. Gah! I can't stress enough how much of a mistake assuming dimensionality is. Everything that Quantum Mechanics does it does despite the assumption of dimensions.
Now... we know that the fundamental constituents of the universe are directed relationships.
Every other concept must emerge from directed relationships. There are no photons, charge, mass or dimensions, as elemental objects. Everything we perceive emerges from directed relationships.
This is the nature of our universe. We know that there are no fixed points. Without a fixed point we cannot define anything. Mathematics and physics should be embracing this knowledge. We don't need to attempt to define anything... because we know we can't.
Our universe is a set of relationships that changes over time. This is fundamental physics.
The only question is how the relationships change over time.
See... physics isn't impenetrable mathematics.
Physics is just a description of how one pattern of relationships becomes the subsequent pattern of relationships.
There's nothing metaphorical about this. The universe is a pattern of relationships that change over time.
Interestingly, we are more interested in the process of change than the specific pattern.
We now have some interesting starting points.
There are infinitely many ways to construct a sequence of states.
That is a lot.
Let us try to narrow it down a bit.
Let us assume a deterministic process. It isn't much of an assumption – the alternative is literally undefinable.
Let us also assume a degree of simplicity. Our investigations suggest localised consistent behaviour...
So, given a pattern of relationships, how can we change that pattern of relationships?
There are no values to be changed. There are no inherent properties of any kind.
A change is, clearly, a change in the pattern.
What is the smallest possible change we can imagine?
Take a point that is related to some other points. Call him A. Let us change A. We are at A. We... ? Delete a relationship... well... fair enough, but as a long term proposition it lacks a certain Je ne sais quoi.
Add a relationship? Add a relationship to what... we are just A. We only know about A. We've got some relationships to other stuff... but we don't actually know about other stuff. We've got nothing with which to construct new relationships. Shuffling relationships here doesn't impact the overall pattern of relationships.
In short... there is a definite minimum change that can be made to any given network. We could imagine some system that deleted and added relationships... somehow. However, we haven't specified a mechanism of deletion and creation of relationships... easier to leave creation questions to our brethren and concentrate on what we can do once created.
Without creation or deletion of relationships, we are left changing the start points of relationships (remember that all relationships come with bias (directionality) built in). Some kind of... swap...?
Technically we should be able to get rid of objects as placeholders altogether. Relationships between relationships is rational... If you like your Category Theory rough then we can eliminate objects as an independent entity altogether.
However, I'm going to appeal to objects as anchors and say that a given change is some rearrangement of relationships across a set of objects.
Our universe could be such that the minimum rearrangement of relationships requires all relationships to determine. That each subsequent state depends on the entirety of the previous state.
But it /looks/ like many small changes, rather than fewer more complex changes, are the rule.
The minimum number of objects for a swap is two.
Ah... but... sequence matters. If we consider the whole system is updated per tick then synchronisation is handled. If a subsection can change independently then we must also specify the pattern of the subchange. That is... there are no gaps for god to sneak in... we either fully define the process of change or don't bother.
Our process of change must specify the actual change, and what the next subject for change will be.
Okay... this physics stuff is really coming together... we just have to find which of the infinite possibilities corresponds to our particular speck...
So... while you are off doing that... let us look at that minimum possible change...
A pair of objects could have any number of relationships between them. Which leaves a corresponding number of permutations of rearrangements.
If we restricted the number of relationships that an object could have we would, naturally, reduce the number of possible permutations.
Okay... every object has zero relationships... yeah... not sure what we can do with that.
Every object has one relationship. Okay... so... we have a ring of objects... umm... not good. There is no way to rearrange a ring into anything other than a ring.
We haven't invented number systems yet... so fractional relationships are definitely too sophisticated for our taste.
So... umm... errr... oh yeah... we could have each object always having two relationships. We would then have 4! possible rearrangements (with one being identity).
Hmm... well... I know that a bunch of objects with one relationship each is a pretty boring ring. I'm not sure if two relationships each and just changing the relationships at two objects at a time is enough to create more interesting behaviour...
Pfft... 24 possible rearrangements (with identity) and some fiddle faddle over which pair to rearrange next... whap it into the computer and let us have a look see...
Really.... whap it into a computer and have a look see.
Oh look... there is some complex behaviour... there is some very complex behaviour...
Gosh... isn't that interesting. Who would have thought that such complexity would be so readily available. Obviously not enough complexity for an actual universe with actual sentient entities.... no... I'm sure our universe is some other much more complex rearrangement... why it would be absurd for the first arbitrary system we construct that shows a bit of interesting behaviour to have any relationship with our actual universe... absurd, I tell you.
No.... it is just an interesting system with some complexity...
…
Cellular Automata had the good graces to include a reference frame. You could see a glider, a shooter, an eater doing their things directly.
There is no right way to view a collection of relationships. It can be pretty tough just telling whether two graphs are actually the same graph or not. (Really Tough!)
There are, however, a few basic measures that we can take for a given graph. Most obviously are the radius and diameter of the graph... the average and maximum path lengths from one element of the graph to all other elements. If we take snapshots of these values as we iterate a graph we might get a rough indication of some underlying behaviour...
But in order to tell whether a given sequence of relationships corresponds to our universe is tricky.
We know that our universe is a sequence of relationship networks. But there are no patterns that are inherently a photon, or an electron, or anything else we observe. There is no 'charge', 'spin' or 'mass' … there is just a sequence of networks. All of the properties we know and love are emergent qualities. Some interaction of changing relationships.
We know that among the properties of complex system is sensitive dependence on initial conditions. In practise this means that it is freaking hard to establish definite correspondences between our real world observation of emergent properties and possible emergent properties within a network of relationships.
In light of this it might be tempting to fall back on Quantum Mechanics. It works, mostly... and who really cares if we actually understand what charge, mass, distance really are... We can do the monkey thing where we repeat the pattern we learned without having the faintest clue WHY the pattern works....
Or we can accept that our tools are there to inform us. They have informed us that the only possible fundamental structure is a network of relationships.
Fortunately for us monkeys... the fruit isn't just low hanging... it is the first thing in front of your face.
Actually seeing it will require someone to realise that for all of Quantum Mechanic's success as a predictive model... it actually tells us nothing about the structure of the universe.
By paying attention to what mathematics is telling us we can gain greater insights into the structure of the universe than provided by a hundred years of very expensive observation.
Following a thing to its logical conclusion shouldn't be perverse for mathematicians.
This is an instance where the beauty of pure thought is magnificent.
We can know, without any doubt, what the structure of the universe is. Because we have been able to rule out all other possibilities. Without a fixed point... the only thing left is a network of relationships.
...
@Forest Goose: You give the impression that repeatedly asking “why” is a bad thing. There is that association with four year olds who repeatedly ask “why”.
Why does the bacteria not grow around that mould? Why is the sky blue? Why does the sun circle the Earth?
All of science is asking “why” (and “how”). Simply accepting the first answer as being good enough leads to religion, flat earths and leeches.
Yes, there are times where you want to just get on and do it. But I can't imagine a scientist thinking that asking “why” should have some definite limit.
@gmalivuk: Presumably you feel that “there are no objective truths” is, itself, an objective truth and therefore paradoxical and therefore disproves itself. So... does that mean you think there are objective truths? Or that there aren't objective truths?
There is no (technical) axiomatic mathematics... yet it was the attempt to construct axiomatic mathematics that showed axiomatic mathematics cannot exist.
Without being able to define axioms, we can't define proof either. We have no logic in a rigorous and absolutely defined sense.
I could have worded it better: “we cannot communicate objective truths”.
@lightvector: It isn't a question of whether axioms are true or not... it is a question of whether they can be stated. As much as you feel that you understand some given set of axioms... you don't in an absolute sense. You probably do have a similar conception for a given set of axioms as most other mathematicians. And for many purposes this sufficient. A subjective sense of meaning is valid and useful when everyone shares a similar subjective sense of meaning.
You are right that there is similarity between the rules of chess and the rules of mathematics. So long as everyone agrees, it doesn't matter what the absolute meaning is. However, in the technical sense of axiomatic mathematics... there is no axiomatic mathematics.
@elasto: I agree with you about the potential of humans. However, humans don't need to avoid axioms... they have absolutely no choice in the matter. Axioms and the logic that depends on axioms don't, and can't, be defined or stated in any degree.
What does exist is the ability to describe patterns consistently. As I have said before... I'm not trying to be nihilistic. The nonexistence of axioms isn't something that has suddenly appeared. They never existed... and yet we have been pretty successful without them. We have been successful despite many people thinking they were doing something that turns out is impossible.
We have hear the difference between perception and action. While people have been thinking that they were doing axiomatic stuff... they have actually been doing something else. Just imagine how much more efficient we could be at becoming gods if we actually understood what we were doing.
Nobody has ever done axiomatic mathematics. It doesn't exist. But we have been doing something with a degree of success. Perhaps we should make an effort to understand what it is we have been doing all this time.
This isn't me making a bizarre argument. This is well understood mathematics. Albeit not well understood by everyone.
Obviously mathematicians are doing something that frequently gets labelled as axiomatic mathematics. This is fine, so long as everyone remembers that what is labelled 'axiomatic mathematics' has absolutely nothing to do with the technical definition of axioms and axiomatic mathematics.
Having said that... it can be really difficult to see past our assumptions. To whit: I made a mistake.
Of course the empty set is an attempt to specify a fixed point.
If you assume counting exists then, bizarrely enough, counting exists. Zero or empty are not supposed to be arbitrary labels. They are supposed to be properties that can be uniquely recognised.
Being able to count the number of relationships is proof positive that you are assuming a fixed point.
Set theorists should know this.
As tempting as it is to think that there is a definite nothing against which something can be counted... just thinking zero means that you have already accepted 1 and 2.
Assume that all objects have an implied self reference and that there is no fundamental way to distinguish between two objects or between two relationships  the same logic that gives us THE empty set rather than AN empty set.
We have a ring of 5 objects A>B>C>D>E>A.
Each object in that ring is linked to the next item in the ring and itself. Start at a point on the ring. Look backwards along the ring. Look forward along the ring.
What would be the difference in your observation if the ring was now six elements long?
Seven? A hundred? A million? Just one single element in the ring?
Without any external reference, there is no way to distinguish between rings of different sizes.
Why do you think set theory fights so hard to prevent identity relationships?
If you assume that you can count the number of relationships that an object has (the number of elements in a set), then you don't need ZFC to justify number systems to you. You've already assumed the existence of counting and an ability to distinguish one number from another number.
The whole point of not being able to define axioms is that you can't define axioms. Being able to identify a particular set by any property whatsoever is presupposing some form of identifier.
Any start point can only prove itself. ZFC gives you number systems because you assume that you can count an empty set.
Set theory knows this. If you permit identity relationships and don't start with the counting system you are supposed to be proving... then... well... you can prove diddly squat.
The contortions that set theory goes through to pretend it isn't making any assumptions would be endearing if it didn't involve so many grownass men.
To be fair... I was buying into the delusion myself. I mean, “empty” is so obvious... so self evident...?
And I absolutely can count up to ten on my fingers and toes.
But axiomatic theory isn't common sense. Axiomatic theory is built upon the idea that knowledge can be manipulated in a deterministic fashion. It is a compelling narrative. It seemed plausible. But in practice it relies on you already assuming whatever it is you are trying to show.
For all the people who seem to think that they already know and understand this... what are those outstanding millennial prizes about?
Mathematics already has the answer. Axiomatic theories cannot be stated or defined in any way.
No ifs, buts or maybes...
What is left is just a matter of consensus. 1+1=2 because, by and large, give or take we agree that it should. There is no logic. There is only what works. And sometimes, things only work because everybody is agreeing that it is working and it gets completed before it fails.
And Uncle Sam is quite definite that any... leeway... in thinking will be in his favour.
I do credit axiomatic theory with being intelligent enough to be able to demonstrate that it was completely the wrong way to approach knowledge.
In the mean time, somebody should really sweep up that four million dollars that they are giving away because they apparently didn't notice what was obvious to all other mathematicians... there is no axiomatic mathematics. At all.
As much as people are justifying it because it works... so did sending children down the mine and up chimneys.
Ahem. Before we get too deep into the “bad”, “old”, “obsolete” maths versus the “shiny”, “good”, “healthier” mathematics... (too late?)
...
We have already learned from that old, fangled, math, malarkey that we can distinguish some things... but if we can't definitely distinguish two things then we are free to consider them the same thing. (Hence THE empty set).
Using set theory with an explicit self relationship for every set and no empty set we find that vast numbers of systems cannot be distinguished. It isn't just simple rings that can't be distinguished from a single point... all symmetric structures are indistinguishable within set theory.
A complete digraph of twenty elements is indistinguishable from a ring of a billion elements (again – within set theory and with no external reference).
We may choose to minimise redundancy and regard all such possible structures as one representative... which tells us nothing about whether there really is just a self reference or actually umpteen bazzillion different elements. We just can't do anything to distinguish between the possibilities so we assume a particular possibility to taste...
This is about what we can communicate directly. It doesn't matter what we think we know... it matters what we can communicate. Without a fixed point, symmetry provides no distinguishing features. Counting the number of relationships requires that we can uniquely distinguish relationships.
So... we need a distinguishing feature. We need an asymmetry.
Imagine one of the absurdly big infinities that set theory invented to make itself feel good... imagine that many points all related to each other. And imagine a single self referential point.
We have no way of distinguishing between these two things (in set theory).
Once we remove the artificial fixed points, set theory cannot distinguish between any symmetric systems. All symmetric systems are indistinguishable from a single set which contains itself (identity).
So, obviously the good stuff is where it isn't symmetric.
As an aspiring god looking to create “not symmetric” stuff it can be a bit difficult working out how to begin.
Our first attempts saw us trying to construct things in a logical fashion... but we couldn't establish a fixed point and without a fixed starting point our actions were of no significance.
Symmetry gets boring real fast since, for all practical purposes, it is no different than identity.
So... lets create us some of the assymmetry... let's see... we just grab this bit and... oh yeah... we can't distinguish that bit from any other bit.... hmmm...
Well... I'll invent a difference... I'll say that A is not B. There!
But B is not A. I can't distinguish between A and B. They appear identical in every respect even though I asserted that they were opposites.
Okay, so we need a third point. Each point balances the other two and isn't directly equal the other.... hang on... three points of an equilateral triangle are indistinguishable...
Dang. Assymetry is more difficult than it looks.
Bugger this for a game of billiards... cheat.
Relationships are asymmetric.
Asymmetry is necessary and exists. Even the saltiest dog needs an occasional bone. The existence of relationships and their directionality are the irreducible components of knowledge. Whether or not justified as axioms... all of knowledge must be representable with these components. All languages function by describing relationships.
There is an argument to be made that the existence of asymmetric relationships is a fixed point upon which we can build. An argument I have considerable sympathy for. Even if we can't describe relationships or asymmetry except by reference to themselves... they still present a (necessary) starting point.
Big shout out to Category Theory for putting the functors up front and recognising that objects are just placeholders between the real action.
We've already seen that all symmetric systems are indistinguishable. We can take this one step further... without a method to uniquely distinguish relationships, a given object cannot be identified by counting the number of relationships. Which leaves us no way to distinguish between sets.
Let us jeer at silly set theories that believed identifying sets was of any relevance to anything (at all).
Yay for Category Theory that focuses on the only stuff we can actually specify... the shape of relationships.
One placeholder is completely indistinguishable from another placeholder. By the same token, one relationship is indistinguishable from another relationship.
Axiomatic logic has taught us to try to understand the placeholders and individual relationships. There is nothing to be gained by trying to understand individual objects or relationships. Their internal structure is permanently outside our ability to perceive or communicate.
So... we know that all of everything is a pattern of relationships.
No... seriously... all of everything is a pattern of relationships.
Physics is a pattern of relationships.
There are no fundamental objects. Trying to define photons, electrons, or any other supposed elementary particle, is to not understand mathematics.
The particle centric view of the universe blinds physicists to properly understanding their subject.
Quantum Mechanics supposes that there is an absolute reference frame... apart from assuming a fixed reference frame... this hides the question of what the stuff between particles is.
Defining particles with respect to dimensions is only useful if you have a complete understanding of dimensions.
And we know that we don't have a complete understanding of dimensions... because that would require that we could define them axiomatically. Really... this no axioms stuff has real world implications.
As successful as Quantum Mechanics is at pattern matching... it tells us nothing about the fundamental nature of the universe.
Space is not just some background against which particles do the fun stuff.
The process that results in what we observe as particular particles interacting is what defines space just as much as it defines the particles.
Explaining things in terms of 'spin', 'charge', 'mass' doesn't explain anything.
As a job of pattern matching... Quantum Mechanics is impressive. As a tool for understanding the universe it is a shit hot mess (that isn't a compliment).
Fundamental physics is as far from common human experience as it is possible to get. It is simply not sane to try to describe fundamental processes in terms of concepts that we know cannot be defined.
At our level of technology our theory of physics must have sufficient fidelity to be credible. We are good enough at pattern matching to find a pattern with a good degree of coincidence with observed reality. As an exercise in mathematics (specifically pattern matching) there is nothing remarkable about Quantum Mechanics. Gah! I can't stress enough how much of a mistake assuming dimensionality is. Everything that Quantum Mechanics does it does despite the assumption of dimensions.
Now... we know that the fundamental constituents of the universe are directed relationships.
Every other concept must emerge from directed relationships. There are no photons, charge, mass or dimensions, as elemental objects. Everything we perceive emerges from directed relationships.
This is the nature of our universe. We know that there are no fixed points. Without a fixed point we cannot define anything. Mathematics and physics should be embracing this knowledge. We don't need to attempt to define anything... because we know we can't.
Our universe is a set of relationships that changes over time. This is fundamental physics.
The only question is how the relationships change over time.
See... physics isn't impenetrable mathematics.
Physics is just a description of how one pattern of relationships becomes the subsequent pattern of relationships.
There's nothing metaphorical about this. The universe is a pattern of relationships that change over time.
Interestingly, we are more interested in the process of change than the specific pattern.
We now have some interesting starting points.
There are infinitely many ways to construct a sequence of states.
That is a lot.
Let us try to narrow it down a bit.
Let us assume a deterministic process. It isn't much of an assumption – the alternative is literally undefinable.
Let us also assume a degree of simplicity. Our investigations suggest localised consistent behaviour...
So, given a pattern of relationships, how can we change that pattern of relationships?
There are no values to be changed. There are no inherent properties of any kind.
A change is, clearly, a change in the pattern.
What is the smallest possible change we can imagine?
Take a point that is related to some other points. Call him A. Let us change A. We are at A. We... ? Delete a relationship... well... fair enough, but as a long term proposition it lacks a certain Je ne sais quoi.
Add a relationship? Add a relationship to what... we are just A. We only know about A. We've got some relationships to other stuff... but we don't actually know about other stuff. We've got nothing with which to construct new relationships. Shuffling relationships here doesn't impact the overall pattern of relationships.
In short... there is a definite minimum change that can be made to any given network. We could imagine some system that deleted and added relationships... somehow. However, we haven't specified a mechanism of deletion and creation of relationships... easier to leave creation questions to our brethren and concentrate on what we can do once created.
Without creation or deletion of relationships, we are left changing the start points of relationships (remember that all relationships come with bias (directionality) built in). Some kind of... swap...?
Technically we should be able to get rid of objects as placeholders altogether. Relationships between relationships is rational... If you like your Category Theory rough then we can eliminate objects as an independent entity altogether.
However, I'm going to appeal to objects as anchors and say that a given change is some rearrangement of relationships across a set of objects.
Our universe could be such that the minimum rearrangement of relationships requires all relationships to determine. That each subsequent state depends on the entirety of the previous state.
But it /looks/ like many small changes, rather than fewer more complex changes, are the rule.
The minimum number of objects for a swap is two.
Ah... but... sequence matters. If we consider the whole system is updated per tick then synchronisation is handled. If a subsection can change independently then we must also specify the pattern of the subchange. That is... there are no gaps for god to sneak in... we either fully define the process of change or don't bother.
Our process of change must specify the actual change, and what the next subject for change will be.
Okay... this physics stuff is really coming together... we just have to find which of the infinite possibilities corresponds to our particular speck...
So... while you are off doing that... let us look at that minimum possible change...
A pair of objects could have any number of relationships between them. Which leaves a corresponding number of permutations of rearrangements.
If we restricted the number of relationships that an object could have we would, naturally, reduce the number of possible permutations.
Okay... every object has zero relationships... yeah... not sure what we can do with that.
Every object has one relationship. Okay... so... we have a ring of objects... umm... not good. There is no way to rearrange a ring into anything other than a ring.
We haven't invented number systems yet... so fractional relationships are definitely too sophisticated for our taste.
So... umm... errr... oh yeah... we could have each object always having two relationships. We would then have 4! possible rearrangements (with one being identity).
Hmm... well... I know that a bunch of objects with one relationship each is a pretty boring ring. I'm not sure if two relationships each and just changing the relationships at two objects at a time is enough to create more interesting behaviour...
Pfft... 24 possible rearrangements (with identity) and some fiddle faddle over which pair to rearrange next... whap it into the computer and let us have a look see...
Really.... whap it into a computer and have a look see.
Oh look... there is some complex behaviour... there is some very complex behaviour...
Gosh... isn't that interesting. Who would have thought that such complexity would be so readily available. Obviously not enough complexity for an actual universe with actual sentient entities.... no... I'm sure our universe is some other much more complex rearrangement... why it would be absurd for the first arbitrary system we construct that shows a bit of interesting behaviour to have any relationship with our actual universe... absurd, I tell you.
No.... it is just an interesting system with some complexity...
…
Cellular Automata had the good graces to include a reference frame. You could see a glider, a shooter, an eater doing their things directly.
There is no right way to view a collection of relationships. It can be pretty tough just telling whether two graphs are actually the same graph or not. (Really Tough!)
There are, however, a few basic measures that we can take for a given graph. Most obviously are the radius and diameter of the graph... the average and maximum path lengths from one element of the graph to all other elements. If we take snapshots of these values as we iterate a graph we might get a rough indication of some underlying behaviour...
But in order to tell whether a given sequence of relationships corresponds to our universe is tricky.
We know that our universe is a sequence of relationship networks. But there are no patterns that are inherently a photon, or an electron, or anything else we observe. There is no 'charge', 'spin' or 'mass' … there is just a sequence of networks. All of the properties we know and love are emergent qualities. Some interaction of changing relationships.
We know that among the properties of complex system is sensitive dependence on initial conditions. In practise this means that it is freaking hard to establish definite correspondences between our real world observation of emergent properties and possible emergent properties within a network of relationships.
In light of this it might be tempting to fall back on Quantum Mechanics. It works, mostly... and who really cares if we actually understand what charge, mass, distance really are... We can do the monkey thing where we repeat the pattern we learned without having the faintest clue WHY the pattern works....
Or we can accept that our tools are there to inform us. They have informed us that the only possible fundamental structure is a network of relationships.
Fortunately for us monkeys... the fruit isn't just low hanging... it is the first thing in front of your face.
Actually seeing it will require someone to realise that for all of Quantum Mechanic's success as a predictive model... it actually tells us nothing about the structure of the universe.
By paying attention to what mathematics is telling us we can gain greater insights into the structure of the universe than provided by a hundred years of very expensive observation.
Following a thing to its logical conclusion shouldn't be perverse for mathematicians.
This is an instance where the beauty of pure thought is magnificent.
We can know, without any doubt, what the structure of the universe is. Because we have been able to rule out all other possibilities. Without a fixed point... the only thing left is a network of relationships.
...
@Forest Goose: You give the impression that repeatedly asking “why” is a bad thing. There is that association with four year olds who repeatedly ask “why”.
Why does the bacteria not grow around that mould? Why is the sky blue? Why does the sun circle the Earth?
All of science is asking “why” (and “how”). Simply accepting the first answer as being good enough leads to religion, flat earths and leeches.
Yes, there are times where you want to just get on and do it. But I can't imagine a scientist thinking that asking “why” should have some definite limit.
@gmalivuk: Presumably you feel that “there are no objective truths” is, itself, an objective truth and therefore paradoxical and therefore disproves itself. So... does that mean you think there are objective truths? Or that there aren't objective truths?
There is no (technical) axiomatic mathematics... yet it was the attempt to construct axiomatic mathematics that showed axiomatic mathematics cannot exist.
Without being able to define axioms, we can't define proof either. We have no logic in a rigorous and absolutely defined sense.
I could have worded it better: “we cannot communicate objective truths”.
@lightvector: It isn't a question of whether axioms are true or not... it is a question of whether they can be stated. As much as you feel that you understand some given set of axioms... you don't in an absolute sense. You probably do have a similar conception for a given set of axioms as most other mathematicians. And for many purposes this sufficient. A subjective sense of meaning is valid and useful when everyone shares a similar subjective sense of meaning.
You are right that there is similarity between the rules of chess and the rules of mathematics. So long as everyone agrees, it doesn't matter what the absolute meaning is. However, in the technical sense of axiomatic mathematics... there is no axiomatic mathematics.
@elasto: I agree with you about the potential of humans. However, humans don't need to avoid axioms... they have absolutely no choice in the matter. Axioms and the logic that depends on axioms don't, and can't, be defined or stated in any degree.
What does exist is the ability to describe patterns consistently. As I have said before... I'm not trying to be nihilistic. The nonexistence of axioms isn't something that has suddenly appeared. They never existed... and yet we have been pretty successful without them. We have been successful despite many people thinking they were doing something that turns out is impossible.
We have hear the difference between perception and action. While people have been thinking that they were doing axiomatic stuff... they have actually been doing something else. Just imagine how much more efficient we could be at becoming gods if we actually understood what we were doing.
Nobody has ever done axiomatic mathematics. It doesn't exist. But we have been doing something with a degree of success. Perhaps we should make an effort to understand what it is we have been doing all this time.

 Posts: 396
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Axiomatic mathematics has no foundation
Treatid, earlier you said that it is possible to unambiguously state tautologies. Do you or do you not stand by that position?
If you do still acknowledge the ability to state tautologies, do you or do you not recognize the possibility that a tautology might be of the form (something)>(something else)?
If you recognize the possibility of such a tautology, do you or do you not recognize the ability to categorize tautologies in a manner such as (something shared by all tautologies in the category)>(something specific to a particular tautology in the category)?
If you recognize the ability to make such categorizations, do you or do you not have any objection to giving a name to the shared left side of that structure within a particular category of tautologies?
If you allow a name to be given to the shared left side of a group of tautologies, what's wrong with the name "axioms of set theory"? It's just a name like any other.
Do you or do you not object to using the term "proof" to describe the act of demonstrating that a particular statement is indeed a tautology?
If you do not object to any particular notion given above, there's no reason to object to "proving" something given certain "axioms", because that's literally just another way of saying that we can demonstrate a tautology that follows a particular format.
If you do still acknowledge the ability to state tautologies, do you or do you not recognize the possibility that a tautology might be of the form (something)>(something else)?
If you recognize the possibility of such a tautology, do you or do you not recognize the ability to categorize tautologies in a manner such as (something shared by all tautologies in the category)>(something specific to a particular tautology in the category)?
If you recognize the ability to make such categorizations, do you or do you not have any objection to giving a name to the shared left side of that structure within a particular category of tautologies?
If you allow a name to be given to the shared left side of a group of tautologies, what's wrong with the name "axioms of set theory"? It's just a name like any other.
Do you or do you not object to using the term "proof" to describe the act of demonstrating that a particular statement is indeed a tautology?
If you do not object to any particular notion given above, there's no reason to object to "proving" something given certain "axioms", because that's literally just another way of saying that we can demonstrate a tautology that follows a particular format.
 Forest Goose
 Posts: 377
 Joined: Sat May 18, 2013 9:27 am UTC
Re: Axiomatic mathematics has no foundation
First, this has, essentially, devolved into raw incoherent ranting at this point, so I'm just responding to what you said to me.
Thanks for the straw man, by the way.
My Point:
If your response to anything is to ask "why", and then "why" to that, ad infinitum: you are no longer actually looking for reasons, you are actively doing the opposite; you are silencing reasoning. That was my entire point, if your theory of knowledge/reasoning/logic/foundations/etc. ultimately ends with "you can't say anything", then you're doing it wrong. The problem with infinite "why"'s is because not being able to answer the question, eventually, does not invalidate any of your previous responses  going back to the ass/chair example, even if I can't tell you the ultimate nature of asses and chairs, I can still assert my ass is in one and both make sense and be correct. I feel like you are doing a very complicated version of this with axiomatic methods, or that that's what led you here.
Three quick suggestions/notes:
1.) Please quit saying things like:
etc.
You have not, at any point, demonstrated this claim, not even kind of. You aren't stating it as a conclusion to a long argument that I disagree with either, you are just stating it like a forgone conclusion, that's just bad form  and ridiculous.
2.) You're all over the place; the reference frames and quantum magic and emergent thises, etc. none of that really seems to connect up with the rest of what you are saying. Maybe it does, I doubt it, but maybe  but if it does, you aren't providing those connections, it just looks like you might be foaming at the mouth.
3.) You haven't successfully argued any single point of your own, please quit with the condescending attitude towards mathematics (a subject you seem to have only a very basic grasp of). I've spent like 20,000 hours of my life studying foundational/computational/logic related branches of math (that's not counting hours towards other branches of math, either). I don't say this as a bragging point, but, rather, that you come off us very insulting  you don't appear to know anything more than a few basic concepts (people have been giving you wikipedia links, etc. on basic things...), so when you are telling me that my whole field doesn't work with a giant rant, it's insulting. Seriously, cut all of the condescending remarks, physics ranting, millennium problem referencing stuff out, you have like two sentences, both are you just stating your conclusion.
To better understand this: imagine going to, say Google, and the only thing you know about computers is how to add binary numbers, then telling them that search engines don't actually exist because of how you think the word "search" should be defined/used and what you think a computer might be. How do you think they would react? You come off the same  and it's almost impossible to take you seriously given that and the way you present your points. Get a little more humble, tighten up your argument, and let the logic of it speak for itself  to be crude: if your posts were a one night stand, it would be 99.9% of the night talking about how good you were in bed, then giving a kiss on the forehead and going home.
@Forest Goose: You give the impression that repeatedly asking “why” is a bad thing. There is that association with four year olds who repeatedly ask “why”.
Why does the bacteria not grow around that mould? Why is the sky blue? Why does the sun circle the Earth?
All of science is asking “why” (and “how”). Simply accepting the first answer as being good enough leads to religion, flat earths and leeches.
Yes, there are times where you want to just get on and do it. But I can't imagine a scientist thinking that asking “why” should have some definite limit.
Thanks for the straw man, by the way.
My Point:
If your response to anything is to ask "why", and then "why" to that, ad infinitum: you are no longer actually looking for reasons, you are actively doing the opposite; you are silencing reasoning. That was my entire point, if your theory of knowledge/reasoning/logic/foundations/etc. ultimately ends with "you can't say anything", then you're doing it wrong. The problem with infinite "why"'s is because not being able to answer the question, eventually, does not invalidate any of your previous responses  going back to the ass/chair example, even if I can't tell you the ultimate nature of asses and chairs, I can still assert my ass is in one and both make sense and be correct. I feel like you are doing a very complicated version of this with axiomatic methods, or that that's what led you here.
Three quick suggestions/notes:
1.) Please quit saying things like:
There is no (technical) axiomatic mathematics
Mathematics already has the answer. Axiomatic theories cannot be stated or defined in any way.
etc.
You have not, at any point, demonstrated this claim, not even kind of. You aren't stating it as a conclusion to a long argument that I disagree with either, you are just stating it like a forgone conclusion, that's just bad form  and ridiculous.
2.) You're all over the place; the reference frames and quantum magic and emergent thises, etc. none of that really seems to connect up with the rest of what you are saying. Maybe it does, I doubt it, but maybe  but if it does, you aren't providing those connections, it just looks like you might be foaming at the mouth.
3.) You haven't successfully argued any single point of your own, please quit with the condescending attitude towards mathematics (a subject you seem to have only a very basic grasp of). I've spent like 20,000 hours of my life studying foundational/computational/logic related branches of math (that's not counting hours towards other branches of math, either). I don't say this as a bragging point, but, rather, that you come off us very insulting  you don't appear to know anything more than a few basic concepts (people have been giving you wikipedia links, etc. on basic things...), so when you are telling me that my whole field doesn't work with a giant rant, it's insulting. Seriously, cut all of the condescending remarks, physics ranting, millennium problem referencing stuff out, you have like two sentences, both are you just stating your conclusion.
To better understand this: imagine going to, say Google, and the only thing you know about computers is how to add binary numbers, then telling them that search engines don't actually exist because of how you think the word "search" should be defined/used and what you think a computer might be. How do you think they would react? You come off the same  and it's almost impossible to take you seriously given that and the way you present your points. Get a little more humble, tighten up your argument, and let the logic of it speak for itself  to be crude: if your posts were a one night stand, it would be 99.9% of the night talking about how good you were in bed, then giving a kiss on the forehead and going home.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
Re: Axiomatic mathematics has no foundation
arbiteroftruth wrote:Treatid, earlier you said that it is possible to unambiguously state tautologies. Do you or do you not stand by that position?
(didn't mean to ignore your last post  was about to edit...)
I don't think I said that. In terms of axiomatic logic... everything is a tautology. As such, the degree of ambiguity is irrelevant.
If you do still acknowledge the ability to state tautologies, do you or do you not recognize the possibility that a tautology might be of the form (something)>(something else)?
Without axiomatic logic, the only thing that a statement proves is itself. Hence I can accept A > A but not A > B.
Having said that... if we start at A: A > B > C >D > A then B, C and D are aspects of A. That is, you can't go anywhere from a given starting point. Whether it was initially apparent or not, B, C and D were all already aspects of A.
Bear in mind that A is still impossible to state or define in any way.
If you recognize the possibility of such a tautology, do you or do you not recognize the ability to categorize tautologies in a manner such as (something shared by all tautologies in the category)>(something specific to a particular tautology in the category)?
The differences between tautologies that I referred to lies solely in the pattern that they form. That is, we can construct patterns of relationships. From an axiomatic perspective, all such patterns are tautologies (even when they don't form closed loops). But we can tell the difference between different patterns. And we do perceive some patterns of relationships as being more significant to us than other patterns.
If you recognize the ability to make such categorizations, do you or do you not have any objection to giving a name to the shared left side of that structure within a particular category of tautologies?
We, as human beings, can categorise things in a subjective manner. What can't be done is to categorise things in an objective manner (axiomatically).
If you allow a name to be given to the shared left side of a group of tautologies, what's wrong with the name "axioms of set theory"? It's just a name like any other.
I have no problems with labels. Technically I don't have a problem with what mathematicians do.
I understand that axioms are not supposed to be true of false or anything else. They are simply an arbitrary starting point. The trouble is that the starting point cannot be defined. The tautology starts right as the axioms are being stated.
Do you or do you not object to using the term "proof" to describe the act of demonstrating that a particular statement is indeed a tautology?
Yeah  we aren't quite on the same page about where the tautology starts.
The idea of axioms is that from the starting point, everything else follows in a consistent fashion. I'm completely happy with this concept. This is what I understand axioms to be.
The trouble is that the starting point cannot be defined in the first place. We cannot specify that set of rules that we are going to follow.
Axioms are stated in a language. But that language itself is not an axiomatic system. We can define ZFC in terms of Category Theory. But ultimately we rely on some natural language to define our first set of axioms. And that natural language isn't defined in the axiomatic sense.
The logic that is supposed to follow from a set of axioms is perfectly reasonable.
The trouble is that mathematicians forget to apply that logic in order to get the axioms in the first place: A statement only has meaning/significance/proof with respect to a set of axioms. So how do you state the axioms if the statements are not also part of an axiomatic system.
Where is the beginning?
If you do not object to any particular notion given above, there's no reason to object to "proving" something given certain "axioms", because that's literally just another way of saying that we can demonstrate a tautology that follows a particular format.
Unfortunately...
The whole problem comes down to defining a fixed point. Axioms are supposed to be the fixed point upon which we can build. But axioms themselves need to be built (defined). And we don't have a first set of axioms from which we can unambiguously build other axioms. It is a chicken and egg problem without evolution to bootstrap us.
arbiteroftruth wrote:The point people have been trying to get across to you is that that's all mathematics tries to do in the first place.
Your careful narrowing down of the options does help to make clear where some of the disagreement lies. Thank you for that.
I'm happy with the idea of axioms. I'm not trying to accuse the deterministic process that is supposed to follow from a set of axioms or being a tautology. IF it were possible to state a set of axioms then I believe it would be possible to construct genuine new insights on top of those axioms.
The problem lies before the deterministic extrapolation from an arbitrary set of rules.
The rules themselves are not axiomatically defined. Even if you offset the problem (define ZFC in Category Theory) you will ultimately arrive at a set of statement that are not themselves axiomatically defined.
It is the starting point that doesn't exist. And without the starting point, there is nothing else to axiomatic theory.
Axioms aren't "fixed points". Axioms are just the stuff we put on the left side of a tautology.
That is what a fixed point it. Or it would be... in actuality... the left side is completely undefined and undefinable. Which leaves the right hand side equally undefined.
We can and do manipulate patterns. And those patterns do have significance for us as humans. But without definite unambiguous axioms as a starting point, there is no axiomatic theory. There is no proof. The manipulations we do are justified because they work for us. There might even be a degree of consistency. But there is no logic (in the axiomatic sense).
You yourself have acknolwedged that tautologies are allowed, and that not all tautologies are the same as each other. So there should be nothing wrong with wondering about tautologies that follow some particular format.
This is accurate... but we know that they cannot follow the axiomatic format. Axiomatic logic requires a known starting position. We have gotten good at reducing that starting point to just a couple of concepts... but axiomatic logic does need a fixed starting point, however small. And we simply don't have that fixed starting point.
That also fits with your position that we should only focus on relationships between things without any notion of what the things themselves are. Just as "A>A" is a tautology no matter what A might happen to be, even if A is completely meaningless, likewise "(axioms about sets)>(theorem about sets)" is a tautology no matter what a "set" might happen to be, even if "set" is a meaningless term.
Sets are a good starting point. I'm happy with sets. Except: The empty set is a fixed starting point. No identity (set containing self) for sets is also a fixed point. The axiom of choice is another attempt as a fixed starting point.
Trouble is... a set theory with no equivalent to the empty set and that allows all forms of recursion tells you nothing.
When mathematicians talk about things being "proven" or about sets "existing" or about things being true given a set of axioms, all that is is a linguistic shorthand for describing which tautologies we find interesting. You may have a point in complaining about overly Platonic attitudes among mathematicians, but that would be a social problem, not a problem in how mathematics itself is structured.
Those platonic attitudes are deeply ingrained. There are several million dollar prizes outstanding based on the idea that axioms are real.
I could accept that this was just an issue of labeling and convenience in expressing ideas... except that that there are significant open problems in mathematics that don't have any justification according to the rules of mathematics.
As noted, this isn't about the axioms we choose being a bit arbitrary. I'm completely fine with the idea of inventing rules and seeing what happens if you follow them. The basic idea of axioms makes sense to me. The problem is that it isn't possible to state a set of rules unambiguously.
There is nothing wrong with subjective truth. Since it is the only type of truth we have access to  I'm actually completely fine with subjective truth.
Axiomatic theory assumes objective truth. It assumes some starting point from which things can be judged to be true (proven). The idea is attractive. But it simply isn't possible to establish that starting point.
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Re: Axiomatic mathematics has no foundation
The XKCD Players
are proud to present
Pachyderm
being an Allegorie
in one ittybitty little act
Dramatis Personae
Dit A. Ert, being the only sane man
Chorus, being Geek
are proud to present
Pachyderm
being an Allegorie
in one ittybitty little act
Dramatis Personae
Dit A. Ert, being the only sane man
Chorus, being Geek
A bare stage, conveying no meaning.
Dit A. Ert wrote:Elephants do not exist!
Chorus wrote:Sure they do, for we have seen them.
Dit A. Ert wrote:Elephants do not exist! I have proven it!
Chorus wrote:Look, an elephant.
Dit A. Ert wrote:It cannot be an elephant, for elephants do not exist!
Chorus wrote:Look, another elephant.
Dit A. Ert wrote:Yes, I see a large gray mammal. But it is not an elephant, since elephants don't exist. This is wellunderstood, but not by anyone but me.
Chorus wrote:But that large gray mammal is an elephant. That's what we mean when we say elephant. And you concede that it exists.
Dit A. Ert wrote:Yes, zoologists may have studied large gray mammals with long, prehensile smellerators and large, flappy air vibration transducerants. But those aren't elephants. They can't be, since elephants do not exist. Let's figure out what those things are, so we can study them.
Chorus wrote:What's a smellerator?
Dit A. Ert wrote:It's a device in the middle of one's face for detecting airborne aromatics.
Chorus wrote:A nose, then. And I bet an "air vibration transducerant" is an ear. What you're describing is an elephant. An elephant is a large gray mammal with a long, prehensile nose and large, flappy ears.
Dit A. Ert wrote:Haven't you been paying attention? Elephants don't exist! I have proven it! Now, let us determine what this strange creature is.
Exeunt.
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 gmalivuk
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Re: Axiomatic mathematics has no foundation
It seems no one has made an animated .gif of an elephant dropping a mic...
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