Is Addition Really a Basic Skill?

[Hurkyl]

Since we're talking about the philosophy of mathematics, I'll move this thread here.


Things have changed slightly since Pythagoras's time. Science/philosophy decides what axioms you should be using, and mathematics tells you what those axioms imply.

 

[NeutronStar]

Science/philosophy decides what axioms you should be using, and mathematics tells you what those axioms imply.

Actually I wish that was true.

 

[matt grime]

Yea waddup Grimey. I've noticed that you realized that i gave a definition. The purpose of the 'definition of red post' was to point out that the person who was asking me to prove red was getting off the topic

that post you refer to wasn't off topic, it perfectly illustrated that just because you can ask for something to be proven, doesn't mean the question makes sense. And when do you get to start using a familiar version of my surname?

 

[matt grime]

Actually I wish that was true.

That is true.

Plus you're logical proposition that you won't prove here is trivially true since the conditional is false.

 

[CrankFan]

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed".

Mathematics isn't intended to be a good model of the quantitative nature of the universe; whatever that means. I think maybe you're confusing the roles of mathematics and physics.

You've not given any justification for your belief that the foundation of mathematics, as it's currently understood, is insufficient or inconsistent. Instead, you've insisted that some facet of the current approach is wrong (like say, defining 0 first) and fail to show how it leads to either a contradiction or limitation in characterizing some useful area of mathematics. Essentially, you don't like construction of N starting from 0, but can't articulate why this is problematic; it's just a (religious) belief.

Regarding Peano, the first publication of the Peano postulates appeared in "Arithmetices Principia", which was published in 1889. In that document 1 is defined as first natural number.

Later, in 1892(?), "Formulario Mathematico" is published, which shows 0 to be the first natural number. Peano changed his approach between those two publications.

But, more to the point: The selection of 0 or 1 as the first natural number is probably only a matter of convenience.

 

[jcsd]

Thanks, I've always wondered why some versions of Peano's axioms start with 0 and others start with 1.

 

[NeutronStar]

You've not given any justification for your belief that the foundation of mathematics, as it's currently understood, is insufficient or inconsistent. Instead, you've insisted that some facet of the current approach is wrong (like say, defining 0 first) and fail to show how it leads to either a contradiction or limitation in characterizing some useful area of mathematics. Essentially, you don't like construction of N starting from 0, but can't articulate why this is problematic; it's just a (religious) belief.

With all due respect it wasn't my intent to prove or show anything. I was merely responding to the original poster's concern with proving addition. I simply providing information concerning that topic.

I think that there's been a general consensus among the responders to this thread that addition cannot be proven (or as some claim it is even meaningless to ask such a question). I agree that it cannot be proven using current mathematical axioms. And I also agree that within that axiomatic framework it is a nonsensical question.

I was merely pointing out that the particular axiomatic framework that is preventing the question from being meaningful was introduced rather recently (with respect to the entire history of mathematics). And that the theory that prevents it is Georg Cantor's empty set theory.

I believe that everything that I've said here is true. I believe this on solid logical grounds. I actually don't have a religious bone in my body. (maybe spiritual, but that's another topic)

I have no intent to attempt to convince people who aren't interested in this topic. The mathematical community as a whole is well aware that there are logical problems associated with set theory. This is no secret and it has been a philosophical debate for many years (not the specific empty set concept that I am referring to, but set theory as a whole logical system).

Just for a quickie I will give you concrete example of the problem,…

Originally (that is to say that before the time of Georg Cantor) a set was intuitively understood as a collection of things. Cantor then introduced the idea of an empty set. A set containing nothing.

Well, this is actually a logical contradiction right here. We have a collection of things that is not the collection of a thing. This is a logical contradiction to the very definition of what it means to be a set. Similar objections actually came up by other mathematicians at the time that Cantor proposed his empty set idea. These objections were never (and I mean NEVER) fully resolved. In fact, the significance of their implications wasn't even fully understood at the time. Nor does it seem to be fully understood today.

In any case, there are only two ways to get around this logical contradiction. One is to claim that nothing is a thing in its own right. Therefore the empty set does indeed contain a thing and there is no logical contradiction. However, that logic leads to further contradictions by the simple fact that they empty set is then no longer empty. It contains this thing called nothing. In fact, this solution was pretty much tossed out as being far too problematic. The so-called "genius" of Cantor's idea was to remove the idea of number from any connection to the idea of a thing thus making it a "pure" concept. Even Cantor did not like the idea of treating nothing as a thing.

This leaves us with the second choice,… simply change our intuitive idea of the notion of a set. A set is no longer considered to be a "collection of things". That is merely an intuitive notion that is not needed for an axiomatic system to work. Instead, Cantor suggested, let's just ignore definitions, and forget about trying to comprehend the idea intuitively and make an axiom that simply states, "There exists an empty set". He somehow sold this idea to the mathematical community and they bought into it.

However, later on (possibly even after Cantor's death, I'm really not sure on that) the mathematical community started seeing other problems cropping up. For example, there must be a distinction between a set, and an element within a set. If this distinction isn't made then 1 would equal 0.

In other words, by Cantor's empty set theory

0 = {}
1={{}}

Now if there is no distinction between an element, and a set containing a single element then there is no difference between Cantor's definition of zero and One.

I'll grant you that this may appear quite trivial but I assure you that it is not.

The problem does not exist for Cantor's higher numbers because they are more complicated combinations of sets and elements.

2 = {{{}},{}} for example. Even if the elements were to stand alone there would be not be equal with any other number. In other words, you can remove the outermost braces (which merely convey the idea of a set) without reducing the contents (the actual collection of things) to configuration that represents a different number.

By the way this is much easier to see if you actually use the symbol err to represent the empty set. Or I think modern mathematicians use the Greek letter phi.

The bottom line to all of this is that Cantor's empty set theory is logically inconsistent. It's based on a logical contradiction of an idea of a collection of things that is not a collection of a thing.

So big deal you might say. It's a trivial thing just let it go and get on with using the axioms. Well if you think like than then you truly are a modern mathematician.

This logical contradiction does exists none the less, and it really does have an effect on the logic that follows from using these logically flawed axioms.

Two things should be apparent right way. First off, there is no formally comprehensible idea of a set. If you think of a set as a collection of things you are wrong. That is an incorrect idea in Cantor's set theory. It simply doesn't hold water in the case of the empty set which is the foundation of the whole theory.

There are far reaching logical consequences to this logical contradiction. And in a very real way they are almost like relativity. Just like relativity is hardly noticeable at small velocities, so the problem with Cantor's empty set theory is hardly noticeable for quantities much less than infinity.

But just as relativity comes into play and speeds that approach the speed of light, so do the logical errors of Cantor's empty set theory come into play at quantities that approach infinity, and more precisely it affects the concept of infinity immensely.

Georg Cantor is the only human ever to start with nothing and end up with more than everything. His set theory leads to ideas of infinities that are larger than infinity. In other words, it leads to the logical contradiction of some endless processes being more endless than others. If that's not a obvious logical contradiction I don’t' know what is, yet this absurd notion has been accepted and embraced by the mathematical community.

Finally, what I have typed into this post is merely the tip of the iceberg. I'm not about to write a book on an Internet forum to try to explain what most mathematicians should already be aware of. But there are other problems associated with the empty set theory as well, and they have to do with counting, or countability.

The whole reason why Cantor introduced the idea of an empty set in the first place was to avoid having to tie the concept of number to the concept of the things that are being quantified. In other words, at the that time in history it was extremely important to the mathematical community that numbers be "pure". I might add that this was much more of a religious notion than a logical one so if we want to claim that someone was being religious rather than logical we should look at the mathematical community around the time that Georg Cantor lived!

In any case, by starting with nothing, Cantor was able to avoid having to deal with the concept of the individuality of the elements (or things) being collected. In other words, by starting with nothing he basically swept the prerequisite notion of the individuality of the objects that are being quantified under the carpet. And by doing so he has removed that constraint from any elements. In other words, in Cantor's set theory anything can be counted as an "individual" element even if it has no property of individuality. This is, in fact, the very reason why he is able to have infinities larger than infinity. He is actually counting objects that have no property of individuality yet he treats them as though they do. I can actually show the error of his ways using his diagonal proof that the irrational numbers are a larger set than the rationals say. Once you understand the whole problem concerning the property of the individuality of the elements it's a fairly obvious proof.

I don’t even know why I'm bothering to type this in actually. The mathematical community simply isn't ripe for this knowledge yet. It just isn't in the "air". I think that it will be soon though as more and more mathematician begin to study group theory where is most likely to become apparent. Some clever mathematician somewhere is bound to realize what's going on and become famous for discovering the "problem".

I think that the most important thing for mathematicians to realize is that this problem was only introduced into mathematics about 200 years ago. Compare that with the age of mathematics and we can basically say that it "just happened!".

I mean, correcting modern set theory will have no affect on things like Pythagorean's theorem, or Euclid's elements, or the vast bulk of mathematics including even calculus which came before Cantor's time. It's main impact with be in group theory. But group theory is becoming extremely important in modern science so it could end up having a big impact there.

Fortunately Cantor's illogical set theory won't affect most normal algebra or other mundane calculations, so don't expect to get tax refund checks from the government when the problem is finally corrected. The government never thinks in terms of sets anyway, they think in terms of bucks.

 

[Gokul43201]

Umm...didn't read all of that, but I believe the question makes no sense not only from an axiomatic point of view, but from a basic, language point of view.

The question makes as much sense as "prove running". This is different from asking "why does running along the (B-A) vector get you to B if you start from A ?", which while self-evident (because that's how it's defined), is at least not an improper usage of language.

 

[arildno]

How do you prove proof by not proving proofs of the principles of proof?
Or something completely different..

 

[CrankFan]

I think that there's been a general consensus among the responders to this thread that addition cannot be proven (or as some claim it is even meaningless to ask such a question).

Right, they're saying that "prove addition" is no more meaningful a statement than "prove purple bonnet banana."

Some people, (like me), have made assumptions about what the OP wanted (probably a mistake ), and sketched out a process by which the function of addition is proven to exist and retain all of the properties we intuitively expect it to have. -- these proof sketches have been ignored by the OP. No one seems to know what he has in mind when he uses the phrase "prove addition", and it's doubtful that it has any sensible meaning.

I was merely pointing out that the particular axiomatic framework that is preventing the question from being meaningful was introduced rather recently (with respect to the entire history of mathematics). And that the theory that prevents it is Georg Cantor's empty set theory.

This is nonsense. Why should the development of some theory prevent a person from asking a meaningful question? Are you claiming that if someone was asked to "prove addition" in 1850 that it would be meaningful then? But now, in 2004 it's not meaningful? and Cantor is to blame for his apparent ability to control people's behavior from beyond the grave?

I believe that everything that I've said here is true. I believe this on solid logical grounds.

Actually you've made many factual errors in your posts, more than I'd bother to correct. It's surprising that you say you believe everything you've said is true, shortly after you've been presented with information about Peano's postulates which contradicts your claim that he was some kind of advocate for starting from 1 as opposed to 0 ... as if any of that mattered.

The mathematical community as a whole is well aware that there are logical problems associated with set theory.

The problems which were widely known at the time of the development of set theory do not exist in modern set theory. If you replace "set theory" in the above quote with "naive set theory" then I don't have a problem with it, but if you mean ZF by "set theory" then your statement is false.

Originally (that is to say that before the time of Georg Cantor) a set was intuitively understood as a collection of things. Cantor then introduced the idea of an empty set. A set containing nothing.

I don't know the origin of the phrase empty set, or the first use of the term set, but I'm a little bit skeptical of taking your claims at face value given the egregious errors you've made previously about the history of mathematics.

I've only studied material which is a refinement of Cantor's work, not the original -- but I don't see how Cantor or his original work is relevant to the supposed problems of set theory as it's understood today.

Well, this is actually a logical contradiction right here.

No it's not.

If one of the axioms asserts that sets are non-empty, then it would be a contradiction for an empty set to exist, but there is no such axiom or theorem of ZF.

We have a collection of things that is not the collection of a thing. This is a logical contradiction to the very definition of what it means to be a set.

Again, the axioms of ZF don't assert that sets are non-empty collections of objects as you are implicitly doing. Someone recently posted a version of Berry's paradox, which "proves" that the naturals are finite. It's a very amusing "proof", the problem is that it relies on the ambiguity of natural language. Your mistake above is similar, in implicitly assuming that a collection is non empty -- of course in a formal system all of these details are made explicit, and it's not a problem.

Similar objections actually came up by other mathematicians at the time that Cantor proposed his empty set idea. These objections were never (and I mean NEVER) fully resolved.

Can you cite these specific objections to his "empty set" theory? You seem to be hung up with the notion of the empty set, but I don't think others, even those skeptical of Cantorian set theory at the of its development had any problems with the empty set. I think the objections of mathematicians like Poincare were related to the concept of infinite sets and had nothing to do with the empty set.

When you talk of "problems never fully resolved", I'm not sure if you're talking about philosophical objections to set theory or the classical antinomies.

If you're talking about the later then all of the classical antinomies were addressed in the development of theories like ZF. It's inaccurate to say that the problems of Naive set theory exist today.

If you're talking about philosophical objections to the notion of infinite sets, well that's fine I suppose that you have an opinion, but these days, philosophical objections of this sort almost always belong to non-mathematicians, and are inconsequential.

However, later on (possibly even after Cantor's death, I'm really not sure on that) the mathematical community started seeing other problems cropping up. For example, there must be a distinction between a set, and an element within a set. If this distinction isn't made then 1 would equal 0.

This is absurd. Every object in ZF is a set, there are no obvious problems with the theory.

In other words, by Cantor's empty set theory

0 = {}
1={{}}

Now if there is no distinction between an element, and a set containing a single element then there is no difference between Cantor's definition of zero and One.

This is false. You're confusing the concepts of set membership and subset.

The empty set {} is a member of {{}}, it is not a member of {}. By extentionality we know that {{}} isn't the same set as {}, since {{}} contains at least 1 element not in {}.

The bottom line to all of this is that Cantor's empty set theory is logically inconsistent.

It's strange how you keep referring to Cantor's "empty set theory".

A quick search at the following site indicates that Cantor isn't known to be the first one to use the phrase empty set or null set.

http://members.aol.com/jeff570/mathword.html

From what source are you determining that Cantor was the originator of the concept of the empty set. Note that I'm not claiming that he didn't, just that I'd like to know if this is true.

I'm not sure if Cantor's set theory can be said to be inconsistent. Certainly not as a result of the nonsensical complains you've made about the empty set. Cantor seemed to be aware of the dangers of unrestricted comprehension, apparently he wrote about it in letters to other mathematicians. That aside; whatever Cantor's theory was (consistent or inconsistent) it has no bearing on the state of set theory today. If you know of a contradiction in ZF, you can state it in the language of ZF and if can't your claim need not be taken seriously.

So big deal you might say. It's a trivial thing just let it go and get on with using the axioms.

It's not that what you've said is trivial, it's that what you've said is incorrect. Your opinions are based on misconceptions. That you have difficulty in distinguishing between elements of a set, and its subsets tells me that you need to go over some introductory material before you're able to discuss these issues competently.

 

[jcsd]

I think Neutron Star one of your main misconceptions is the between naive set theory and ZF set theory. naive set theory which is the set theory of Cantor's day the empty set is not axiomatic as you are allowed to define sets by shared properties of their mebers and clealy if you define the shared properties of the members of a set in such a way that it cannot have any members you get the empty set (for example {x|x is a nonegative real number less than zero} defines the empty set). However defining sets in such a way leads to paradoxes such as Russell's paradox.

ZF set theory remedies this by saying that a set is only defined by it's members, in ZF set theory the empty set is axiomatic and as far as I am aware no paradoxes arise because of it's inclusion.

Secindly this i sgetting off track as Peano's constructio of the nautarl numbers are not depedent on whethr or not the empty set exists (it is not an orginal idea to notice that sets are defined by ther members yet the empty set has no members, it has been pointed out before, but no logical inconsistenty arises because of this state of affairs).

 

[NeutronStar]

I think Neutron Star one of your main misconceptions is the between naive set theory and ZF set theory.

I'm not actually having any misconceptions between naive set theory and ZF. In fact, I see them as totally separate things.

ZF may very well be a logically sound theory in its own right. But that is totally irrelevant.

I am only concerned with the following conditional statement:

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed".

So even if ZF is perfectly logically correct, if it doesn't correctly model the quantitative nature of our universe then it doesn't really matter that it's logically sound. It's still incorrect within the context that should matter to a scientist.

I never mentioned a word about ZF because ZF did indeed toss out the idea of a set as a collection of things. That's how ZF got around the logical contradictions of Cantor original set theory. And yes, that is a way around it but at what price? In ZF the notion of a set is unbound, and undefined. This leads to paradoxes like infinities that are larger than infinities.

But wait! Hey, that's only a paradox if we think of infinity in a comprehensible way as an endless process or quantity. Only then is it a paradox. But if we are willing to forfeit comprehension of ideas then we can accept that abstract infinities can be larger than other infinities. Of course it no longer makes sense to comprehend infinity as a simple idea of endlessness because it makes no sense to have something that is more endless than something else.

So ZF is only a sound logical theory for those who are willing to give up comprehension of ideas and see no paradoxes in things like collections of things that contain no thing, or ideas of endlessness that are more endless than other ideas of endlessness.

To get rid of the paradoxes all we have to do is forfeit our comprehension of the ideas. Seems easy enough. But for me it just isn't something that I'm willing to do.

Moreover I don't see any need to do it. I can see a logical and sound set theory that is actually based on a comprehensible idea of a set as a collection of individual things where the property of individuality has also been defined in a comprehensible way. We can still have a symbol for the absence of a set which is yet another very comprehensible idea. The set containing all positive numbers less than zero is absence, it simply doesn't exist. But we can use a symbol to denote that it doesn't exist just like we currently use a symbol for the non-comprehensible idea of an empty set.

What's of much more importance is that once we build a set theory on this foundation we will quickly see that all sets cannot be elements of other sets because all sets do not have a valid (by definition) property of individuality. Therefore we won't be bothered by Russel's paradox concerning the set of all possible sets because such an idea is an illegal idea by definition.

So far we are doing just as good as ZF.

But much more importantly there is so much more that falls out of this idea,…

To begin with it doesn't lead to infinities larger than infinity. There can only be one condition of endlessness, a set either has this property or it doesn't. No paradox there. It's also quite comprehensible as simply an idea of endlessness. So we drop that paradox off as well.

We do however pick up a lot of new interesting stuff that I'm also afraid to mention. (ha ha)

One consequence of this "new" set theory is that the number point in a finite line must necessarily be finite. At first that might seem hard to swallow, but actually it makes perfect sense after it is understood why this must be so. It become crystal clear by a very simple proof of why it must be so. At first I had a lot of problems accept this myself, but after thinking about it for many years I have come to grips with why it must be so. I also think that it is amazing that a corrected definition of the idea of number can actually lead to something that is actually true about the real universe (that it is quantized).

Another thing that comes out of this formalism is that it makes absolutely no sense at all to talk about negative numbers in the absolute sense. Once again, this may come as a shock and seem rather weird, but after thinking about it for a while it makes perfect sense.

There can be no such thing as an absolute negative number. The whole property of negativity is a relative property. This is actually the true nature of the universe we live in. It's a good model of the quantitative nature of reality. Ironically mathematicians are already aware of the absolute properties of number. They even have the absolute value function.

Well if you actually stop and think about it, negativity in any comprehensible sense must always be relative. It makes no sense to talk about an absolute negative quantity. Yet we insist on giving negative numbers a life of their own. The fact of the mater is that negativity is always a relative property between sets. It's not an absolute property of either set. The idea of something being negative must always be in context of a larger picture in order to be comprehensible. Even on a number line negative numbers are only negative relative to the origin of the line. Take away that relative reference and negativity is meaningless.

Much of this falls out of a corrected definition of number (where again, by corrected I mean a definition that genuinely reflects the true quantitative nature of our universe). Once the definition is corrected and set theory properly reflects the quantitative nature of the universe in which we live many things fall out of it including the very quantized nature of the universe. In short, had mathematicians been on the ball they could have predicted the quantitative nature of the universe before Max Planck discoverer it in quantum mechanics. Although, I don't believe that mathematics could have actually put a number on it. But just saying that it must be so would have been quite an achievement.

In any case, I seriously don't care anymore. Call me a nut. Claim that I'm full of errors. I disagree with that of course, but I really don't care if other people believe that. My rambles certainly would be full of errors if compared to ZF because I'm not talking about ZF,…. duh?

I'm talking about my conditional statement:

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed" or at least incorrect within that context.

I really should have known better that to even mention any of this on a pure mathematics site. Forget I ever mentioned it. I'm not out to convert anyone. Honest. I really don't care what anyone believes. Have a nice life and enjoy whatever you choose to believe.

 

[Hurkyl]

NeutronStar, I'll start with a simple question: do you reject the number zero?

 

[NeutronStar]

NeutronStar, I'll start with a simple question: do you reject the number zero?

Yes. If a set is considered to be a collection of thing, and number is defined as the quantitative property of a set, then Zero, by definition, cannot be a concept of number.

It can however be a valid mathematical concept representing the absence of number (or absence of a quantity, or set). This is, in fact, how it is actually used. Take the number 1060 for example. This number means actually represents 1 thousand, no hundreds, 6 tens, and no ones. In other words, the symbol Zero is representing the asbsense of quantity. It is a valid concept of quantity even though it doesn't officially qualify as a number by definition.

In practice the symbol and concept of Zero would be used pretty much in the same way that it is used today. The main difference would be in the actual way that it is comprehended. It would be seen as an absence of quantity, or a set, rather than as a set that contains no elements. Be definition in the strictest formalism it would be understood that it cannot be thought of as a number.

Of course we always knew that Zero was a weird number anyway because of the problems associated with division by the number Zero.

So, yes, I recognize the quantiative concept. But no, technically it wouldn't satisfy the definition of a number, and therefore it would be incorrect to claim that is is a number. It can still be used as a valid symbol of communication as a quantitative idea however.

 

[dekoi]

Addition is only a way we, as human beings, articulate ideas which we discover.

Since what we discover is perfect in every respect (although not always the way we articulate it), then addition itself should also be perfect.

Proving an idea is, like someone else said, proving a metaphysical entitity (or a color as an example).

We can of course, empirically prove addition by the use of several objects (e.g. with apples, like children learn) -- however, that is proving the way we articulate addition, not addition in its own self.

 

[Hurkyl]

I'm sure you see the analogy between the empty set and zero.

While I shudder at labelling it the "absence of blah", I'll go with it for the sake of argument... zero is used when you would normally want to express a quantity, but you have the absense of quantity... the empty set is used when you would normally want to express a "collection", but you have the absense of a collection.

And just like zero, virtually all of the operations applicable to sets can be extended to apply to the empty set.


I don't really think there's any argument that the empty set is any less legitimate than zero would be. The empty set certainly has practical value, just look at any programming language: there would be an absolutely nightmarirish semantics problem if collections were required to have at least one element!


I want to reemphasize an earlier point: just like zero, virtually all of the operations applicable to sets can be extended to apply to the empty set.

There's the old adage, if it looks like a duck, and it sounds like a duck...

Your argument sounds like it's entirely an issue of semantics. To you, a collection must have at least one element, so you have the separate concepts of "collection" and "absense of collection". Just what is the problem of having one term that includes both of these concepts?

For example, you don't reject the term "fruit" just because there are "apples" and "oranges", do you?

You are fixated on the "definition": "a set is a collection of elements". That definition is geared for those who consider the empty collection a collection. For those like you who reject the idea of an empty collection, you should take the "definition" of set to be "a set is either a collection of elements, or the absense of a collection".


Again, just like zero, this unified concept has proven its usefulness: it's silly to reject it, especially over an issue of semantics.


Oh, and a disclaimer: everything in this post is on the philosophy of mathematics -- don't try to take it as the actual thing.

I'll touch on the infinite (notice I did not say "infinity"!) in the next post.

 

[Hurkyl]

Actually, there are several different aspects of your argument, each of which I'd like to address separately. The next is axiomization.

The whole reason why Cantor introduced the idea of an empty set in the first place was to avoid having to tie the concept of number to the concept of the things that are being quantified. In other words, at the that time in history it was extremely important to the mathematical community that numbers be "pure". I might add that this was much more of a religious notion than a logical one so if we want to claim that someone was being religious rather than logical we should look at the mathematical community around the time that Georg Cantor lived!

I offer a third possibility -- it was a pragmatic notion.

You say we should look at the mathematical community of the time. Well, let's look at the early 1800s: the field of analysis was blossoming at a rapid pace, directed mostly by intuition, rather than rigor. Abel had this to say in 1826:

There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has lead to so few of the so-called paradoxes.

One particularly interesting fact is that there was a great reluctance to accept Fourier's method of expanding functions as a trigonometric series (today a very important tool), because it was just too weird. Because there was no axiomization of analysis, Fourier could not prove his method worked.


As you can see, it was a practical necessity that led to the search for a "pure" way to define the numbers, so that results could be rigorously justified, in the spirit of Euclid.

 

[NeutronStar]

As you can see, it was a practical necessity that led to the search for a "pure" way to define the numbers, so that results could be rigorously justified, in the spirit of Euclid.

Well, I'm not sold on the idea that it was a practical necessity. I particularly believe this for the case of Euclid's Elements. While I admire Euclid's genius and totally agree with his logical structure of geometry, I genuinely don't see the need for the axiomatic method. He could have done the same thing by defining intuitively comprehensible concepts. I believe that because I can deconstruct Euclid's axioms and reconstruct them into intuitively comprehensible concepts without having to change any of his rules whatsoever. So from my point of view the mere fact that he chose to present it as an axiomatic system is totally irrelevant to me.

I do understand why he chose to use axioms rather than definitions. It's simply easier to state an axiom than to have to explain a definition to the point where everyone will comprehend it in the same way and agree on its meaning. So there is a practical aspect to axiomatic methods.

I'm actually not against axiomatic methods. I could actually state my intuitive set theory as a set of axioms also. There would be absolutely no problem with doing that. In fact, as far as I can tell any comprehensible logical system should be able to be stated in an axiomatic format.

What I do have a problem with is when a logical system is stated in axiomatic format and no one can explain it in terms of intuitively compressible ideas. That's when I step off the boat. I like things that I can comprehend. I don't like to have to learn a bunch of rules that I don't fully understand, and that lead to paradoxes that I can't even comprehend. That makes no sense to me. My first intuition is to reduce the axioms to comprehensible ideas, understand them, then put them back into axiomatic form and continue. With set theory I wasn't able to do that because the axioms don't make intuitive sense to me.

This really bothered the hell out of me for years. I struggled to make logical sense of them for literally decades. I finally concluded that they simply are nonsensical and this is why I can't make any sense of them. Then I decided to see if I could build a system that does make sense and I could. All I have to do is start with Peano's original idea of Unity (although I like to call it the property of individuality), and then define the number one on that. In essence I am defining an element first, and then I'm going to go on and define a set based on the concept of collections of elements. There can be no empty set in my theory because the whole concept of the set is based on the concept of a collection of an element. If we have no element, we can't have a set.

Ok, so it's an intuitively comprehensible theory. So what? Does that make it dangerous?

It can be stated axiomatically. Actually I never really thought about stating it axiomatically, but what would stop me from doing so. It's based on totally comprehensible ideas so all I need to do is state those idea as axioms.

I would start with the axiom that there exists an element. Then I would go on to state the axioms that define its necessary property of individuality. The property of individuality is the crucial difference between my set theory and Cantor's. Those properties. I actually have a workable axiom for that is based on the element's definition of existence. In a very real sense it is based on an operation of subtraction. Or better yet, I should say that it actually defines an operation I call subtraction. Subtraction within this primitive context cannot produce negative sets. However, as the logical system builds the concept of a negative set does come into play. That concept is explained in a relative context. In my system it is clear that there are two entirely different meanings to the negative symbol. One meaning is the relative negativity of a set with respect to a larger picture. I think of that negative sign as an adjective in the language of mathematics. The other meaning of the negative sign is to perform the operation of subtraction, in that context the symbol represents a verb in the language of mathematics.

We actually already recognize these two different meanings of the negative sign in mathematics, but few people actually think of one as an adjective describing a relative situation and the other as a verb describing the action of an operation.

In any case, I won't bore you with any more. The whole idea for me is that it works, and it works in a fully comprehensible fashion that leaves nothing undefined. Moreover, it's actually a more accurate description of the true quantitative nature of our universe. It doesn't lead to logical contradictions such as collections of things that are not collections of things. Nor does it end up leading to infinities larger than infinity, or the logical contradiction of some endless processes being more endless than other endless processes. To me that whole idea is totally absurd. Fortunately my system does not allow that. It turned out to be an unexpected consequence that I had not planned on at all.

There are other things about my system that many people might have trouble with at first. Like the idea that my system leads to the inevitable conclusion that a finite line must only contain a finite number of points. I found that hard to buy into myself at first. But after studying it for years I see now why it must be the case. It turns out that there is a very beautiful eloquent proof using this set theory and the concept of functions (which my set theory does not affect at all) that not only proves that this must be the case, but the proof itself makes it crystal clear why it must be the case. It also makes the idea very intuitively comprehensible.

Oh well, I'm spending far too much time typing this stuff in. I have other work to do. But my point is that I have no problem with axiomatic systems that make sense. I just have a problem with ones that don't make intuitive sense. Oh, by the way, using my set theory one can "prove addition". Or at least explain it in a very logical intuitive way as to clearly show why it must be the way that it is. This comes partly from that property of unity or individuality that current set theory has refused to address. Once that property is included as part of the theory addition can be proven. In other words, it can be proven that addition is a valid operation within the overall logical system. The operation of addition "falls out" of the basic concepts (or axioms) rather than being a part of them. It was actually that fact that attracted me to this thread to begin with.

But now I'm almost sorry that I responded because I really don't have time for this right now.

I supposed I've explained my position far enough. I'm really not prepared to attempt to explain my entire formalism. If I were going to do that I'd write a book about it and get paid for my time.

I'm not a mathematician. I'm a physicist. I want to get back to studying group theory and differential calculus. Unfortunately I need to know these things in order to do physics.

By the way, while mathematicians may have dropped the ball with set theory I will give them credit for doing a wonderful job with the calculus!

 

[alexkerhead]

I reckon addition is a basic skill. But can it be proved?
How?

DO this first.
Look up the definition of "addition"
then use the definition with your fingers.
take the five on your left hand, and add it to the sum of fingers on your right hand. If you are normal, you should find a sum of 10(ten) now, put 6 in the in the first varible, you have 11 then, that is a false statement based on the definition of addition..
You really don't need to worry about addition being provable, becuase you use it in the tense that we define it as.

I hope this helps.

 

[Hurkyl]

I've argued that the empty set is a legitimate concept, and I've argued the practicality of using the empty set. Now I will argue that rejecting the empty set leads to a defective set theory.


One of the more powerful features of sets is that they are able to describe hierarchies. For example, if I'm playing the towers of hanoi puzzle (but, suppose I only have the 7 discs, I don't have the base and pegs), any particular position is a multiset of sets. For example, if I have discs 1, 3, and 5 where the first peg should be, disks 2, 4, 6 where the second peg should be, and disk 7 where the third peg should be, my position is then given by:

[ {1, 3, 5}, {2, 4, 6}, {7} ]

As an aside, this demonstrates the importance of distinguishing between a set and its elements -- if the set {1, 3, 5} meant the same thing as its elements 1, 3, 5, then I lose the entire structure: I have no way of representing the fact that my collection if discs has been partitioned into three distinct parts.


Back to the main point: suppose I then move discs 2, 4, and 6 onto disc 7. Using a definition of set that permits empty sets, I can now describe the position as:

< {1, 3, 5}, {}, {2, 4, 6, 7} >

But without the concept of the empty set, I cannot accurately express this -- I have no way of specifying that there is a third place that could hold a disc, but currently isn't. Because of this deficiency, I argue that a set theory without an empty set is a defective set theory.

(p.s. the reason I said that I need a multiset of sets is to describe the solved position: < {}, {}, {1, 2, 3, 4, 5, 6, 7} >. You need a multiset to express the idea that I have two copies of the empty set)



I don't see what's incomprehensible about the ZFC axioms, except possibly the axiom of choice. (Unless, of course, you steadfastly refuse to try and internalize the notion of an empty set). They are, in "English" rather than formalism:

Two sets are equal iff they have the same elements.

For any A and B, {A, B} is a set.

If I have a set of sets, their union is also a set.

If I have a set, there is a set containing all of its subsets.

There is an empty set. (this is actually superfluous -- the next axiom shows a set exists, and the axiom after that proves the empty set exists)

There is a set of natural numbers.

If I have a set S, and a property, then there is a set containing everything in S with that property.

If {a, b, c, ...} is a set in the domain of some function f, then {f(a), f(b), f(c), ...} is a set.

A set can't contain itself, either directly or indirectly.

And the axiom of choice.


What part of it don't you understand?



And finally, you're beginning to sound awfully crackpottish:

The whole idea for me is that it works, and it works in a fully comprehensible fashion that leaves nothing undefined.

Balderdash. You, for example, never defined the terms "collection" or "element". It's an elementary fact that, unless you're using circular "definitions" you must leave something undefined.

Moreover, it's actually a more accurate description of the true quantitative nature of our universe.

This is another typical crackpot claim -- but it's only "more accurate" in capturing the way you wish to describe things. You have not, for example, performed any measurement of anything and gotten a more accurate answer than a measurement based on a theory that accepts the empty set.


I won't complain yet about "infinity" because I haven't written my response yet on that concept.

 

[matt grime]

I'm not actually having any misconceptions between naive set theory and ZF. In fact, I see them as totally separate things.

ZF may very well be a logically sound theory in its own right. But that is totally irrelevant.

then stop saying modern mathematics is flawed because of a theory that was rejected long ago.

I am only concerned with the following conditional statement:

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed".

But who says the maths is what you think it is ?
I am a professional mathematician, paid to do research, none of what I do has the slightest bearing on the quantitative nature of the universe.

So even if ZF is perfectly logically correct, if it doesn't correctly model the quantitative nature of our universe then it doesn't really matter that it's logically sound. It's still incorrect within the context that should matter to a scientist.

I never mentioned a word about ZF because ZF did indeed toss out the idea of a set as a collection of things. That's how ZF got around the logical contradictions of Cantor original set theory. And yes, that is a way around it but at what price? In ZF the notion of a set is unbound, and undefined. This leads to paradoxes like infinities that are larger than infinities.

And so the lack of comprehension of set theory starts again. That statement is entirely false:

Cantor's notions of different infinite cardinals predates the Zermelo Frankel set theory axioms and is exist in naive set theory too. I have no need to presume my model is a model of ZF in order to demonstrate there is no bijection between N and P(N) using unrestricted comprehension as my rule for set membership.

Moreover, Skolem showed that there is a model of ZF in which every infinite set is countable (there is only one infinity, in your terms).

So are you unaware of these things or are you being deliberately antagonistic?

 

[NeutronStar]

stop saying modern mathematics is flawed because of a theory that was rejected long ago.

I actually put it into a conditional statement to show where I am coming from. I still believe very strongly in the truth of this statement.

IF "mathematics is suppose to be a correct model of the quantitative nature of our universe" THEN "mathematics is incorrect"

…none of what I do has the slightest bearing on the quantitative nature of the universe.

I personally doubt that very much. Just because you may not be consciously aware of the connection doesn't mean that it doesn't exist. I believe that if you are making any reference at all to the concept of number then there is necessarily a connection between what you are doing and the quantitative nature of the universe.

I firmly believe that any idea of number that cannot be reduced to an idea of quantity is necessarily a wrong idea. Why do I believe this? Because, for me, this is the root of the concept of number. We learned the concept of number at a very early age to be the idea of a property of a collection of individual things. This is in fact, the intuitive idea of number that we all use in everyday life. This is the nature of the concept of number.

If through the ages we have lost sight of that basic fundamental concept of number and the word number now refers to some other idea I would very much like to have someone explain to me a what that other idea is in a very intuitively comprehensible way. If they can't do that then I claim that they have no comprehensible idea at all.

What good is an idea that no one can comprehend? It's meaningless.

I have yet to run into a true number that cannot be reduced to an idea of quantity in as a property of a collection of individual things. Where, things may be quite abstract, but yet they can be shown to clearly have a property of individuality.

Now, it is true that I have run into mathematician using numerals as though they are numbers when in fact they aren't. This is a common practice actually. If I name my dog 43 is that a number? I think not. It's simply a label made up of numerals that we normally use to represent the concept of quantity. There is no quantitative property of 43 associated with my dog's name. Now if I have 43 dogs, and name them by number then the 43 can actually have a quantitative comprehensible meaning. Or maybe it's the 43rd dog that I've owned. Once again there is a quantitative context. But if I just arbitrarily name the dog "43", then that symbol has absolutely no connection with the concept of the number 43.

Numerals are not numbers. They are merely the symbols that we use to label the numbers. If you are actually working with numbers, and not just arbitrary meaningless numerals, then you must be working with quantitative ideas. Those ideas came from the observation that our universe exhibits a dependable quantitative nature. (i.e. collections of well-defined individual things combine together in dependably predicable ways) .

If you're numbers don't represent this basic concept then I question whether they actually represent the concept of number at all. What concept do they represent if not collections of individual things? Do you know? Can you explain that concept in a comprehensible way?

I have yet to meet a "meaningful" number that I couldn't explain as a collection of individual "things". Keep in mind that "things" can be quite abstract notions, all they need to possesses is a clearly defined property of individuality. They don’t need to be tangible objects. Without that clearly defined property they cannot be quantized. In other words, if you can't clearly say how many elements you have in a set, then you can hardly put a number on it. The concept of number loses its meaning.

So if your working with numbers that don't represent concepts of a collection of things then you're working with concepts that aren't really numbers at all.

You are fixated on the "definition": "a set is a collection of elements". That definition is geared for those who consider the empty collection a collection. For those like you who reject the idea of an empty collection, you should take the "definition" of set to be "a set is either a collection of elements, or the absense of a collection".

Yes, I am fixated on a comprehensible idea of number as a collection of individual things. It works for me in every case. I have yet to find a number that I cannot comprehend in this way. In fact, by giving each number this test is has helped me to understand what they actually represent. It has also revealed situations where people are using meaningless numerals thinking that they are actually numbers, like in the case of the dog named "43". The 43 is not actually a number in that context. It has no numerical meaning. It's just numerals being used as an arbitrary label.

My concern is not actually with the empty set, but rather with its consequence. By starting with an empty set as a foundation and defining the number one based on that all we are really doing is copping out on the real issue of addressing the concept of individuality. This cop out has actually worked. That is to say that elements in sets in modern set theory do not need to pass any test for their property of individuality. This is what allows us to count things that have no property of individuality thus leading to absurd results like infinite collections that are more infinite than other collections. This all comes from our lack of addressing the individuality of the elements within sets.

NOTE TO EVERYONE

I really didn't come here to be called a crackpot and antagonist.

There's no antagonism on my part. Current set theory is incorrect within the context of my conditional statement at the top of this post. I hold that this is the truth. I will hold that it is true until the day I die because I firmly believe that I have clearly discovered the truth of this.

Call me what you will. Disagree with me all you want. I know that I'm correct in my conditional statement above.

However, since I am being viewed as a crackpot and antagonist I think it's time to move on. No sense in preaching to a hostile audience. What have I got to gain from that? I didn't come here to push my theory. I simply came to point out to the original poster of this thread that current mathematics cannot prove his concerns, and I tried to explain why that is so. Then I attempted to also offer an explanation of why current mathematics can't prove his concerns, and how it can be repaired to the point where it can provide him with an answer.

It is my personal belief that pure mathematicians have lost sight of the origins of their discipline. Mathematics is definitely not a science because it does not conform to the scientific method. Any mathematician who claims that mathematics is a science is clearly wrong, yet I see them do this all the time. However, it is quite possible to design a mathematical formalism that is based on the scientific method. That is pretty much what I have proposed. Such a system is viable, and would be more phenomenological correct. Scientists should be concerned with just how accurate our language of quantity describes the actual phenomenological nature of our universe. After all, this is really the only thing that sciences uses the language of mathematics for. Why not make it phenomenologically accurate?

 

[jcsd]

I'm not actually having any misconceptions between naive set theory and ZF. In fact, I see them as totally separate things.

That's the problem though, you ARE having some major misconceptions about many areas of math and this is clear from your posts.

You have a choice you can either hold onto your misconceptions or find out why you are wrong.

 

[Hurkyl]

How does the equality relation not capture the idea of individuality? I suspect you'll have to elaborate on what that means to you and why it's relevant.



If you're arguing that mathematics has moved away from asking "what is it?" and moved towards "what can I do with it?" and "how can I represent it?", I would certainly agree, and argue wholeheartedly it's for the better.

 

[NeutronStar]

You have a choice you can either hold onto your misconceptions or find out why you are wrong.

But I'm not wrong.

I actually use mathematics everyday in probably very much the same way that you do. I don't have a problem understanding mathematics from a technical point of view. I can take derivatives, or do integration, etc, etc, etc, with the best of them. I just ignore set theory when I'm doing them. I think that most people do this. I also interpret things differently.

For example, if you and I both calculated the volume and surface area of Gabrial's Horn we'd both come up with the same results. The horn would have a finite volume and an infinite surface area.

But then if we were both asked how much paint it would take to paint the horn I'd know how much paint we need and you wouldn't.

That's because we would both interpret the results differently.

If you're arguing that mathematics has moved away from asking "what is it?" and moved towards "what can I do with it?" and "how can I represent it?", I would certainly agree, and argue wholeheartedly it's for the better.

Actually I agree that practicing mathematicians should be concerned with "What can I do with it?" and "How can I represent it". And most of you fellows are indeed practicing mathematicians so that's what you are focusing on. That's actually good.

But the question of "What is it?", or better yet, "Has it been properly defined?" are questions that should be asked by mathematicians that are developing the framework for an entire formalism (like set theory)

Everything that I have been talking about has concerned the actual defintions and rules of set theory (not how to go about using it)

But I totally agree, that once that's been done correctly then the practicing mathematicians shouldn't need to worry about having to deal with primitive concepts on a daily basis.

For example, even when we do calculus like say, taking a limit. Who thinks in terms of the formal defintion of the limit? Nobody. That's a done deal. They are more concerned with proving things about the quantities that they are studying like does it converge, diverge, etc. They don't need to go back an revist the definition of the limit each and every time they use the concept. Neither would they need to go back and revist the concepts of set theory each time they use a number.

I'm just saying that the foundation of set theory is flawed (again within the context of the conditional statement that I gave earlier). Once that's been done (and the affected rules of operations corrected) then it's a done deal and practicing mathematicians go back to doing pretty much what they've always done. The only difference is that they would veiw things differently and some results *may* change.

To be honest, I haven't found any major significant differenences *yet*. Hell, if I had I would be cashing it in for a Nobel Prize instead of sitting here typing this in.

However, I believe that such a trophy exists. Making this change in set theory will have a significant affect somewhere down the road. I believe that it will show up in the field of Group Theory actually. I wish I was more educated in that particular field.

But for the most part with normal calculations all it would really come down to is pretty much a change in intuitive comprehension of the ideas we already have.

I meantioned before, it would kind of be like relativity. It would go almost completely unnoticed for most of normal mathematics, only when things get close to the concept of infinity does it really come into play. Like in the Gabrial's Horn problem.

This really isn't a problem for practicing mathematicians. It's a problem for the more philosophical mathematicians that are concerned with the actual meaning of mathematical statements. I guess that's what philosophy is all about. But in a very real sense that's what science is all about too. When did mathematics depart from philosophy and science to such a large degree that are no longer concerned with intuitive comprehension of their concepts?

And more to the point, why should scientists rely on a mathematics that ignores intuitive comprehension when this is precisely what scientists are seeking to discover?

What scientist wouldn't give his soul to fully understand quantum mechanics intuitively? Why settle for axioms that we don't understand? Why not describe things in an way that we can understand them?

 

[selfAdjoint]

But then if we were both asked how much paint it would take to paint the horn I'd know how much paint we need and you wouldn't.

Why wouldn't he?

 

[NeutronStar]

Why wouldn't he?

Because he was probably a good student of modern mathematics.

 

[matt grime]

As far as I can tell, Neutron, you're an undergraduate or beginning graduate in physics, and you're telling me that my research must be about quantitative things and that I'm wrong?

I'm dealing with homotopy colimits in arbitrary triangulated categories. There is nothing in it that has any bearing on the real world and has any relation to the "number concept", ie quantity.

It is important that my underlying field is countable and algebraically closed - you may try constructing such a thing someday to see why it has little to do with the number concept as you understand it.

Of course this is all based upon unspecified meanings (by all parties) of what they mean by quantitative, isn't it?

Mathematics evidently isn't what you think it ought to be, what ever it is (I know few people who are prepared to say what it *is* precisely, only what it does, and occasionally what it isn't).

I called you antagonistic because you were making repeated and egregious errors that anyone could have checked were factually false, and you weren't admitting them. I don't mean the philosophical ones here but the simple facts such as you asserting that

ZF axioms create the multitude of infinities.

That patently is false by Skolem's paradox.

The different cardinalities of set theory arose *before* ZF, Cantor wrote his main works on this in the late 1800's I believe, ZF was formalized in the 20's, and I think the reference for Frankel these days is a paper in 1930.

Just like the original poster who thought that Cantor's paradoxes were somehow a product of ZF set theory you've got it all the wrong way round.

Irrespective of whether you think that maths is this or that, you've made many errors that you've failed to correct, and have insisted, based upon these plainly wrong ideas, that we aren't able to deal with the real world effectively.

In your proposition, the burden is upon you to justify why all mathematics must deal with the quantitative nature of the universe. Quantity might be one aspect of its nature, but even then you need to justify the conculsion too. All you've done is list a set of untrue assumptions you have of what others think about mathematics.

 

[NeutronStar]

As far as I can tell, Neutron, you're an undergraduate or beginning graduate in physics, and you're telling me that my research must be about quantitative things and that I'm wrong?

Actually I'm retired.

When I said that you were wrong I mean that if you are working with ideas of number and you think that they don't represent quantitative ideas then you simply must be wrong, because it is my belief that ever idea of number can be reduced to a quantitative idea. In fact, I hold that if it can't be reduced as such then it is an idea other than number.

Also, I'm not actually saying that you are wrong in the sense that you actually looked into this and came up with the wrong conclusion. I'm willing to bet that you never even tried to justify your concepts of number as quantitative ideas because, for you, it simply isn't relevant. In fact, it may well be completely irrelevant from your perspective. But it is my belief that someone could reduce your work to quantitative ideas if they choose to look at it that way, and I hold that his must necessarily be the case if your work is associated with ideas of numbers.

So it's not like you actually went out of your way to attempt to see your work in that way and then reported back that it can't be done. You just claim off the top of your head that it isn't necessary to view it that way and therefore it's irrelevant. I disagree.

I'm dealing with homotopy colimits in arbitrary triangulated categories. There is nothing in it that has any bearing on the real world and has any relation to the "number concept", ie quantity.

It sounds like an interesting study. But again, can you actually say that someone actually tried to explain it in terms of quantities and has proven that it can't be explained as such a way? I would find that extremely coincidental had someone actually performed such a study on your work.

It is important that my underlying field is countable and algebraically closed - you may try constructing such a thing someday to see why it has little to do with the number concept as you understand it.

Well, being countable and algebraically closed are certainly ideas related to the behavior of quantity so I highly suspect that if these things are important to your work that your work can indeed be reduced to quantitative ideas whether you want to believe that or not.

Of course this is all based upon unspecified meanings (by all parties) of what they mean by quantitative, isn't it?

I will grant you that. My meaning of quantitative is based on a comprehensible idea of collections of individual objects that can be shown to possesses a clear property of individuality. It is this last property of individuality that we may differ on I think.

Mathematics evidently isn't what you think it ought to be, what ever it is (I know few people who are prepared to say what it *is* precisely, only what it does, and occasionally what it isn't).

Well, my biggest problem with mathematics is that it is beginning to embrace logical formalism that are based on non-quantitative ideas. I see nothing wrong with these non-quantitative logical formalism, but way call them mathematics?

Take Boolean Algebra for example. Many people will say that this is part of mathematics. But why? It certainly has absolutely nothing to do with the idea of quantity. It's a totally different kind of logic. Now binary arithmetic is certainly a quantitative idea and belong to mathematics. But Boolean Algebra is not. Why confuse the issue? Why does mathematics need to embrace every possible logical scheme whether it has to do with ideas of number or not? That makes no sense. Also what then becomes the distinction between the word, mathematics and the word logic. Pretty soon any logical formalism will be under the umbrella of mathematics and we won't need a separate work called logic. To put that another way, mathematics will have been watered down to just become another word for logic.

I called you antagonistic because you were making repeated and egregious errors that anyone could have checked were factually false, and you weren't admitting them. I don't mean the philosophical ones here but the simple facts such as you asserting that

ZF axioms create the multitude of infinities.

That patently is false by Skolem's paradox.

The different cardinalities of set theory arose *before* ZF, Cantor wrote his main works on this in the late 1800's I believe, ZF was formalized in the 20's, and I think the reference for Frankel these days is a paper in 1930.

Just like the original poster who thought that Cantor's paradoxes were somehow a product of ZF set theory you've got it all the wrong way round.

I don't believe that I've made any errors. I've simply been misunderstood. I don't have anything the wrong way round. I'm well aware that Cantor's work came first and Zermelo and Fraenkel were influenced by that. In fact, I believe that I've even said as much somewhere in a previous post. I have looked a the ZF axiom and I see them as nothing more than formal band-aids attempting to fix up all of the logical inconsistencies of Cantor's original work. They were attempting to fix up Cantor's work, not reject it and start over. And that's precisely what they did. They put official axiomatic band-aids on Cantor's original set theory. That's all they did. They didn't correct the theory IMHO.

In fact, I usually refer to their work as the ZF Band-aid

Irrespective of whether you think that maths is this or that, you've made many errors that you've failed to correct, and have insisted, based upon these plainly wrong ideas, that we aren't able to deal with the real world effectively.

You see them as errors. I see them as misunderstandings.

In your proposition, the burden is upon you to justify why all mathematics must deal with the quantitative nature of the universe. Quantity might be one aspect of its nature, but even then you need to justify the conculsion too. All you've done is list a set of untrue assumptions you have of what others think about mathematics.

My bottom line is that any mathematics (or more correctly, any idea of number) that cannot be reduced to an idea of quantity and comprehended as a collection of individual things is simply an idea that departs from the idea that mathematics originally evolved from.

If there is more than one idea of number, then the mathematical community should be able to describe it in a comprehensible way. In my 55 years of life I have never been offered such a description, and I've spent at least 20 years of my life in college classrooms as either a student or an instructor.

If someone could describe these other ideas of number in a comprehensible way I would be quite happy to hear it. But all I get is a bunch of incompressible mush with the word abstract being used, not to imply that their idea of number can be applied to many cases, but rather to imply that their idea of number is vague and incomprehensible in an intuitive sense.

Well, all I have to say is that if someone's idea of number is vague and incomprehensible in an intuitive sense then they have an incomprehensible idea. In other words, they have no workable idea at all.

So if anyone has an intuitively comprehensible idea of number that cannot be reduced to an idea of quantity in terms of a collection of individual things, I'm all ears. I would actually treasure such an idea.

 

[Hurkyl]

You say "comprehensible" and "incomprehensible" quite a lot. Have you ever stopped to think that some people may comprehend things you do not?

To me, it appears your incomprehension seems to arise more from your desire not to comprehend than any inherent incomprehensability in the ideas.

 

[Philocrat]

I reckon addition is a basic skill. But can it be proved?
How?

Your question should have been:


HOW DOES OUR NATURAL LANGUAGE (NL) QUANTIFY THE STATES OF THE WORLD?

and as soon as you ask this question, even without any attempt to answer it, the next question that must result from this is:


HOW DOES NL ACCOUNT FOR THINGS IN DECLARATORY AND JUDGEMENTAL CLAIMS OR PROPOSITIONS?

I personally do not think that there is any need for proof, however, if you insist, NL has only one formula for doing so and that is:

y = x - 1 * x + 1

or equivalently:

y = x * x - 2

I must warn you that this does not necessarily proof anything in the formal mathematical sense. Rather, it merely accurately predicts and states the quantitative and logical structure of NL. So whenever you have a discourse with your peers, every sentence that comes out of your mouth and from your peers' mouth naturally and effortlessly quantifies the states of your world using the above single formua. Infact, this formula remains formally consistent even when you are only merely or unconsciously assuming the number of things involved or quantified in each of your statement. I am therefore suggesting that what you are trying to proof (addition) is already naturally and effortlessly taken care of in the basic quantitative and logical structure of NL.

 

[StatusX]

philocrat: what are you talking about? are you like that guy who kept talking about the triangle inequality? don't you realize that what you said makes no sense?

 

[Philocrat]

philocrat: what are you talking about? are you like that guy who kept talking about the triangle inequality? don't you realize that what you said makes no sense?

Yes, I am aware of that...it could very well be all nonesense. This is the very mistake that science makes all the time. If I say to a scientist how do things add up in the following statements;

1) Einstein is tall

2) Kate kisses anderson

3) My patient has influenza

They will think that I am talking about formal proof. The question now is: in what way do these sentences succesfully quantify the states of the world? It is our mouths that manufacture these sentences with the intention of conveying information from A to B. How do they succeed in doing this...convey the intended information to the perceiver in a manner that avoids errors, vagueness and misunderstading? Well, that is the question. I am claiming that the formula that I provided above formally quantifies the state of the world in every human uterance, even when we are talking vaguely or making references to phantoms.

In fact this quanto-logical form of NL concerns science more than anyone else. When things range numerically beyond observational range, science starts to talk about things in terms of fluxes, waves, energy and all that. Scientists do not count atomic particles on a one to one basis because they do not follow linear and predictable pathways. Hence science starts to talk about moving particlces in terms of fluxes and approximations. Well, I am suggesting that whether you can count atomic particles or not the formula that I provided above at the level of NL is not affected by your inability to do so.

Given that there is a nano-machine that can track every moving particle on its pathway (curved, linear or random) and successfully follow and count every single particle on its pathway, the native speaker of NL who is supplied with this information cannot avoid speaking about these succesfully tracked and counted particle in a manner that fully repsects the quantitative structure of NL. The Formula in NL will quantify and make judgements about those fully tracked and counted particles naturally.

 

[drcrabs]

Your question should have been:


HOW DOES OUR NATURAL LANGUAGE (NL) QUANTIFY THE STATES OF THE WORLD?

You are obviously right and i give you my uttermost apology.
I am sooo sorry.
What was i thinking?
I mean how could i have just blatantly neglected your feelings and intelligence like that?
I don't know how i am going to get to sleep at night.
Once again very sorry.
From now on all questions I ask, no, all thoughts I have, will always respect your superiority and unquestionable intelligence as I do not want to embarrass myself again.
But more importantly, I do not want to embarrass you my asking such a stupidly ludicrous question.
I hope you can forgive me for my unacceptable behaviour

 

[matt grime]

One last one for Neutron. Seeing as I deal with objects that are too large to be sets in any model of ZF*, then I think I can fairly easily claim that quantity isn't that important. Your notion of quantity appears to be that it is something associated to sets, their "size", others may disagree with you there, or at least point out that quantity is a bad word to use when talking about sets that are not finite. It is ambiguous and leads to all kinds of problems with this naive association. How does one even begin to construct cardinals of infinite sets without using the very set theory you despise? (I'm presuming that by calling it a band aid that you don't approve of it.)




* for a proof of this non-quantitative nature can I sugges Neeman, Triangulated Categories, Princeton University Press.

Note that just because I use "quantity" doesn't mean I actually prove anything about it necessarily.

And if I can study something independently of the "quantity", then by the very definition of necessity, I can categorically state that quantity *is* irrelevant. You may argue that using "quantity" might improve the work or shed new light on it, but it certainly isn't necessary. And I would argue that by making no unnecessary assumptions I am doing something good.


As to why finite fields don't have a quantity in the sense I think most people would assume it to mean (ie not that it is a set, more that it deals specifically with the properties of the set of natural numbers): quantity would imply that 1<2, 2<3 and so on, but at some point n==0, and we get n-1<0 if we do it naively. This assumption is based upon the fact that you responded in a thread about addition in the integers, and the ensuing debate about numbers as sets, and that you reject the notion of transfinite cardinals.

If you're saying that the field has a quantity because it is a set, then I cannot argue, since it is true. But as I say, I deal with categories, I deal with things that are not presumed to be sets, and indeed can be shown not to be sets - some homological functor, and not very pathological at that, sends the trivial object to one whose subobjects do not form a set.

 

[NeutronStar]

On Transfinite Qualities

I've been dying to respond to Hurkyl's comments about "comprehension" and the "incomprehensible" because I have some very meaningful and serious point to make on that topic. Unfortunately my time is limited and I'll have to put that on hold for a bit. Hopefully I'll be able to find time to come back to it because I feel that it is an extremely relevant topic.

In the meantime, I feel even more compelled to address some of Matt's comments because he is actually right on target.

One last one for Neutron. Seeing as I deal with objects that are too large to be sets in any model of ZF*, then I think I can fairly easily claim that quantity isn't that important. Your notion of quantity appears to be that it is something associated to sets, their "size", others may disagree with you there, or at least point out that quantity is a bad word to use when talking about sets that are not finite. It is ambiguous and leads to all kinds of problems with this naive association. How does one even begin to construct cardinals of infinite sets without using the very set theory you despise? (I'm presuming that by calling it a band aid that you don't approve of it.)

I actually agree with everything that you said here 100%

The main problem here is that you simply don't know me. You think that I'm just complaining about ZF because I don't like it, and that I have a very limited ability to comprehend ideas, yada, yada, yada. I don't blame you for having that initial impression of me at all because of the fact that you don't know me. However, I assure you that such an impression is far from the truth.

Yes, I do have a bone to pick with blind acceptance of ZF overall. And I do see it as a band-aid for conceptual problems that were never properly address. I see ZF as a formal way of sweeping these issues under the carpet where they will continue to be ignored for as long as ZF is so widely accepted and defended. So I see ZF as a huge rock blocking the path to a better understanding of the universe. It's nothing personal I assure you.

In any case getting back to the point. Yes I agree with you that sets have other properties besides their quantitative properties. I never really meant to imply that they don't. However, I do assert that mathematics historically began as a way to describe the quantitative nature of the universe that is exhibited by the property of sets that we have come to know as quantity. This is what gave rise to our original intuitive understanding of the nature of number. Or perhaps it would be better to stay that this is what gave rise to our intuitive understanding of ideas of quantity that we have come to know as numbers.

This intuitive idea has its basis in the idea of collections of individual things. This property of the individuality of the things in the collection has pretty much been taken for granted. Even ZF doesn't officially address it. It just kind of assumes that it exists informally.

By ignoring this all-important concept we've actually missed the whole point of the concept of number. After all, if the things in a set have no property of individuality then how can assign the set a quantitative property? It would be impossible. A set with the property of 3, for example, implies that it contains 3 occurrences of individual things. If there was no way to verify that this is the case then what would it mean to say that the set has a quantitative property of 3? It would be meaningless.

I'm saying that this all-important property of individuality has been formally ignored by modern set theory. Georg Cantor is the one who tossed this baby out with the bath water when he introduced the concept of an empty set. This was his way of avoiding having to deal with it or formally define this property of individuality. Actually, to be fair to Cantor, he actually felt that this would be a valid way to define the property of individuality. However, if you look at what he is doing closely you will see that he is not defining a individuality that is based on the concept of quantity, but rather he has defined a concept of individuality based on quality. That really is a significant difference when it's all said and done.

Let me see if I can give a more concrete example of what I'm talking about,…

And if I can study something independently of the "quantity", then by the very definition of necessity, I can categorically state that quantity *is* irrelevant. You may argue that using "quantity" might improve the work or shed new light on it, but it certainly isn't necessary. And I would argue that by making no unnecessary assumptions I am doing something good.

Ok, first let me see if I can clear up some possible miscommunication here. I have no problem with studying things that are independent of ideas of "quantity". However, if mathematics is a formalism was initially built from concepts of quantity, and you are studying non-quantitative ideas, the why use mathematics?

Well, I think the reason is obvious. Mathematics has come to be viewed as something more than a just a model of quantitative concepts. But exactly what makes us think that we should be able to do that? It originally arose from our observation that the universe behaves quantitatively, and this is what mathematics basically describes. I like to think of mathematics as a formal language that we invented to describe the quantitative nature of the universe. Because if you think about it, this is historically what it is.

So what makes us think that this logical language has any value when referring to non-quantitative ideas? I think that part of the reason actually has to do with the fact that many mathematical idea don't have any obvious connection to ideas of quantity. However, I hold that in most cases the connection actually exists if we stop and think about it long enough, it just isn't obvious sometimes. But the connection is still there.

Now in your work, you claim that there is no connection. And now that you have explained your work in more detail I actually agree with you. However you aren't working with concepts of numbers. You are working with concepts of transfinite numbers, and that's a totally different ball game!

As to why finite fields don't have a quantity in the sense I think most people would assume it to mean (ie not that it is a set, more that it deals specifically with the properties of the set of natural numbers): quantity would imply that 1<2, 2<3 and so on, but at some point n==0, and we get n-1<0 if we do it naively. This assumption is based upon the fact that you responded in a thread about addition in the integers, and the ensuing debate about numbers as sets, and that you reject the notion of transfinite cardinals.

This again comes down to more of a problem of poor (or an insufficient amount) of communication. We've only been discussing this topic on an Internet board along with other ideas that may serve to contaminant or distract from specific issues.

Yes,… I do denounce the idea of transfinite cardinality as quantitative properties of sets.

No,… I do NOT denounce the idea of transfinite cardinality in general as qualitative property of sets. In fact, I recognize it completely and actually use it to support my view in many ways.

In fact, the whole idea of infinity and how it is manifest itself in different ways is of great interest to me. I simply don't see that as a quantitative concept and question whether the formalism of mathematics is the correct formalism with which to describe it correctly.

I'll give one very quick and crude example here then I must run off, I'm spending far too much time here than I should be actually. I guess I just have no discipline!

Consider the following "three" sets, N, Q, & R. (Natural, Rationals, and , Reals)

Can we actually quantify these sets by colleting them into yet another set say, X.

If we do then what is the quantitative property of X? Well, it's pretty obvious that it must be 3 right? I mean, we collected "three" sets into yet another collection called X, therefore X must have a quantitative property of 3.

Well, this actually isn't true.

Why not?

Because N, Q and R do not have "quantitative" properties of individuality. Therefore, it makes no sense to treat them as individual elements.

But they do, we might argue! They are obviously individual things. They have individual properties. Yes, that's true. But their property of individuality does not manifest itself in a quantitative sense. It manifests itself in a "qualitative" sense that has absolutely nothing at all do to with ideas of quantity. (well that might be going a little bit overboard in this simple case actually, but I would at least argue as much and then listen intently to opposing views on that topic)

In any case, once we recognize that from the point of view of cardinality alone, N and Q represent the same thing, (that is to say that from a purely quantitative point of view there is no distinction between N and Q), then we begin to realize that "quantitative" property of X may actually be 2 and not 3 after all. So what at first appeared to be a quantity of 3 from a naïve quantitative point of view has now become a quantity of 2 from a more qualitative point of view.

Moreover, we begin to realize that we are now quantifying qualities rather than quantities. In other words, we are moving away from a quantitative form of logical reasoning into an area of logic that is based on qualitative reasoning.

Well, I really must go because I'm spending too much time here. But my point is that once we enter into the study of objects that possesses transfinite properties we are leaving the realm of quantitative analysis and moving into a realm of qualitative analysis. That's fine. That's a valid move. But what I am suggesting is that we are erroneously bringing the logic of mathematical formalism (that has actually constructed from ideas of quantity) into our formal analysis of a completely different concept of quality.

Now obviously it works to some extent. But I hold two things.

1. It is actually holding us back from making faster progress.
2. It is actually preventing use from fully comprehending these new qualitative concepts.

It's preventing use from fully comprehending these new qualitative concepts because we aren't looking at it as a brand new formalism. We are looking backward to mathematics at the old quantitative ideas and attempting to apply them to these new qualitative ideas.

I believe that the universe does indeed exhibit other qualitative property besides it's obvious quantitative property. And those properties are important, useful and of interest to study. I just think that we'd do so much better if we realize that mathematics was originally formulated to describe the quantitative nature of the universe and now these new transfinite properties of the universe deserve to have their own logical formalism focusing on them. But as long as we keep the attitude that mathematics applies to EVERYTHING we aren't going to make any progress in that area.

If I ever find time to reply to Hurkyl's comments about the "comprehensible" and "incomprehensible" I'll have more to say about this distinction between quantity and other types of qualities in the universe. For now I simply must get on to other work.

If you're saying that the field has a quantity because it is a set, then I cannot argue, since it is true. But as I say, I deal with categories, I deal with things that are not presumed to be sets, and indeed can be shown not to be sets - some homological functor, and not very pathological at that, sends the trivial object to one whose subobjects do not form a set.

This sounds interesting to me, but I'm afraid I don't fully understand what you said in this quote and I don't have time to think about it right now.

I do believe that fields can be understood via idea of quantity to some degree and they are very useful in that way. However, they may also possesses other types of qualities that cannot be expressed or understood in terms of quantity. If that's the case then wouldn't it be nice to know just how these other qualities might be comprehended intuitively if at all possible?

 

[Hurkyl]

And I do see it as a band-aid for conceptual problems that were never properly address.

The point of ZF was to address the logical problems with Cantor's set theory. It didn't address what you call conceptual problems because they were not considered problems.

I see ZF as a huge rock blocking the path to a better understanding of the universe.

I really think you need to take a step back and reevaluate things. Even if you stick to the spirit of your objections, shouldn't your problem be with the general notion of abstract formalism?

I do assert that mathematics historically began as a way to describe the quantitative nature of the universe

I would disagree. You can't neglect the importance of geometry in mathematics. It's fairly clear that, aside from simply counting, that the notion of number was essentially defined by geometry.


(note: I'm taking my best guess as what you mean by individuality)

This property of the individuality of the things in the collection has pretty much been taken for granted. Even ZF doesn't officially address it. It just kind of assumes that it exists informally.

Of course not; ZF is a set theory. If you want to know about the individuality of points, you turn to geometry. If you want to know about the individuality of integers, you turn to number theory.

he is not defining a individuality that is based on the concept of quantity, but rather he has defined a concept of individuality based on quality.

And what's wrong with that? How can one have a concept of quantity without first having a concept of quality?

it's pretty obvious that it must be 3 right?

...

has now become a quantity of 2

Let's analyze what you have done. You started with one concept of "individuality" -- presumably with respect to some algebraic property (e.g. N, Q, and R are all nonisomorphic monoids). With respect to this concept you have three distinct objects, so the set {N, Q, R} has cardinality 3.

Now, you switched to a different concept of "individuality", where cardinality was the only relevant property. With respect to this concept, N = Q, so the set {N, Q, R} has cardinality 2.


You bring this up as if it is some sort of defect, but it is certainly a common thing, even in natural language. If I ask "how many coins do you have in your pocket?" I would normally expect a different than "how many types of coins do you have in your pocket?"


Now, I have been doing my best not to actually do any math in this thread, but I think something needs to be said about this example: you really cannot be this vague about things. When you're talking about cardinalities you need to say so. {|N|, |Q|, |R|} is indeed a set of cardinality 2 because, as you mention, |N|=|Q|. {N, Q, R} is clearly a different sort of beast than {|N|, |Q|, |R|}.

 

[NeutronStar]

The point of ZF was to address the logical problems with Cantor's set theory. It didn't address what you call conceptual problems because they were not considered problems.

Agreed. And this is why I hold that ZF is a theory that cannot be conceptualized. In other words, it's incomprehensible on a conceptual level.

Whether it's logically sound at this point is moot.

I really think you need to take a step back and reevaluate things. Even if you stick to the spirit of your objections, shouldn't your problem be with the general notion of abstract formalism?

That a very difficult question to answer without a discussion on the semantics of the word "abstract". We may very well use that word to mean quite different things.

Just for a quickie because I have limited time I'll offer the following.

Abstract - A concept that can apply to many cases.

I absolutely support the general notion of abstract formalisms in this respect.

Abstract - A non-tangible object of pure thought.

Again, I have absolutely no problem with abstract concepts in this regard either. In fact, any formalism that can't deal with this would be pretty much useless.

Abstract - Unclear, ill-defined, vague or incomprehensible.

In truth I don't even agree with this definition of the word "abstract" but many people seem to take it to mean this. I totally denounce any formalism that is "abstract" by this meaning of the word. Ironically, if ZF is not concern with maintaining conceptuality then it certainly appears to fall into this category. So if this is what people mean when they say that ZF is an "abstract" theory then I say that it's pretty much worthless.

I would disagree. You can't neglect the importance of geometry in mathematics. It's fairly clear that, aside from simply counting, that the notion of number was essentially defined by geometry.

I don't neglect the importance of geometry in mathematics. I absolutely see it as an extension of the idea of quantity. After all where would geometry be without the idea of "units" and "coordinates"? Both of those are clearly quantitative ideas.

I do however disagree with the idea of a continuum. I don't see it as a necessity to geometry. I also believe that a finite line can only contain a finite number of points. But again, this doesn't really affect geometry. Finally, I believe that irrational quantities don't satisfy a conceptual idea of quantity. But once again, I don't see this as being a problem for geometry.

Numbers like Pi and all the other irrationals can simply be viewed as relationships that can't take place in reality. We can still calculate them to as many places as we so desire. Ultimately we will have to round them off eventually. Or we can create symbols like we already do to represent them if we wish to commutate the ideas precisely. But the point is that those quantities can't really exist in our universe. Pi doesn't actually exists in the universe. There are no perfect circles for example. Ultimately the universe rounds these quantities off at the quantum level.

I have no problem with geometry except its failure to recognize that a finite line can only contain a finite number of points. But that's not really a problem with geometry, that was given to geometry by the mathematics that preceded it.[/quote]

You bring this up as if it is some sort of defect, but it is certainly a common thing, even in natural language. If I ask "how many coins do you have in your pocket?" I would normally expect a different than "how many types of coins do you have in your pocket?"

YES YES YES!

Now, I have been doing my best not to actually do any math in this thread, but I think something needs to be said about this example: you really cannot be this vague about things. When you're talking about cardinalities you need to say so. {|N|, |Q|, |R|} is indeed a set of cardinality 2 because, as you mention, |N|=|Q|. {N, Q, R} is clearly a different sort of beast than {|N|, |Q|, |R|}.

Again, YES YES YES!

But now why is |N|=|Q| and neither of those equal to |R|?

Can you tell me?

 

[CrankFan]

But now why is |N|=|Q| and neither of those equal to |R|?

A bijection can be shown to exist between N and Q, and no bijection exists between N (or Q) and R.

It's extremely arrogant of you to say things like

"Yes, I do have a bone to pick with blind acceptance of ZF overall."

When you're the one who is showing ignorance of the things that you're blindly objecting to.

 

[matt grime]

Mathematics has invented one more thing that you don't appear to think it possesses: linear spaces which deals with "geometry" of a finite number of points.


Your posts seem to belie that idea that you're a platonist: that mathematics is about things that have some existential form. Mathematicians in general do not tend to adopt that philosophy. In fact the very abstraction that you think is holding us back is exactly what allows us to go forward faster. I've apparently convinced you that I'm not doing quantitative things in mathematics or thinking quantitatively, so why do you still state that mathematics is being held back by the (non-existent) dependence we have on quantity? Please also understand that although mathematicians may explain that cardinality is about quantity to the layman, that isn't what we actually think it is. And if you think that adopting Cantor's notion of cardinality has held us back, can you think of a better way of demonstrating there are real numbers that are not computible, or that there exist transcendental numbers? Or any of the many important aspects of probability theory?

Further, if geometry is a quantitative thing, ie coordinates and lengths, then if I have an idealized right angle triangle of sides 1,1 and sqrt(2), and sqrt(2) is irrational, then how come irrational numbers don't represent quantity?

 

[Hurkyl]

Short response this time.

After all where would geometry be without the idea of "units" and "coordinates"?

I'm not sure what "unit" has to do with anything.

And it got along fine for a very long time without coordinates...

And don't forget the work of Tarski that has shown that every question you can ask in the (first-order) language of geometry can be answered either affirmatively or negatively using just the axioms of geometry.

(which, for instance, are not powerful enough to define the notion of integer)


Furthermore, coordinates have attained a reputation for obscuring geometrical facts and ideas (and I agree). They're good for computation, but not for understanding.

I do however disagree with the idea of a continuum. I don't see it as a necessity to geometry.

Maybe that's because you aren't familiar with some of the problems with the alternatives.

You mentioned that you have a problem with irrational numbers. Well, there is an affine plane whose coordinates are rational numbers. But it suffers from the problem that lines can pass through the center of a circle without intersecting the circle.

Another problem shows up with functions. I'm sure you're familiar with the intermediate value theorem:

Thm: If a < b, f is continuous on [a, b], f(a) > 0, and f(b) < 0, then there exists an c in (a, b) such that f(c) = 0.

The thing is, when you don't live in a continuum, then the intermediate value theorem can fail.

 

[Hurkyl]

Oh, two more points.

When I ask "how many coins do you have in your pocket?" I've lost information. I really need to find out "what coins do you have?"

And here's another important thing to consider... you can still answer my question (either one) even if your wallet is empty.

 

[NeutronStar]

A bijection can be shown to exist between N and Q, and no bijection exists between N (or Q) and R.

Bijection? Can you explain what that means conceptually.

Or, of much more importance to this discussion, can you explain conceptually the "qualitative" differences of these cardinalities? Is it a quantitative idea (i.e. Is it believed that one of these sets actually contains MORE elements than the others?). Or is there some other quality in play here? And if so can you describe what it is?

It's extremely arrogant of you to say things like,... {snip},...When you're the one who is showing ignorance of the things that you're blindly objecting to.

With all due respect I'm not "blindly" objecting to anything. I have very sound conceptual reasons for everything that I object to. Just as my questions above imply. Can you explain what you think you know in conceptual terms? In other words, do you really "understand" what you know? Or are you just blindly satisfied that rules that you don't really understand conceptually have somehow been satisfied?

This isn't intended to be an arrogant question. I'm simply seeking the truth. Can you explain what the difference is between the cardinalities of those sets in a conceptual way that any person on the street can understand? If not I sincerely question whether you actually understand it yourself. It's not a matter of arrogance. If you can't explain your ideas in comprehensible terms that can be conceptualized then I have no choice but to conclude that you really don't understand them yourself. This has absolutely nothing to do with personal arrogance. It's simply a matter of logic. You either know what you're talking about or you don't. Don't take is as a personal judgment.

Please also understand that although mathematicians may explain that cardinality is about quantity to the layman, that isn't what we actually think it is.

Well, just like I asked CrankFan. May I please have an explanation of these other concepts of cardinality?

I would like to have them in conceptual terms of course. I mean, after all they are "concepts" are they not? If they can't be explained in conceptual terms I would argue that it is improper to call them "concepts". Mathematicians should not be permitted to say that cardinality represents non-quantitative concepts and then refuse to describe what these other concept are. That's pretty ignorant if you ask me. It just goes to show that even they have no clue.

All I'm asking is for an explanation (in conceptual terms, not just a bunch of axioms) that describe these other qualities of cardinality. If that can't be done, then I have no choice but to conclude that mathematicians have no clue what they are working with. This isn't a matter of being arrogant. It simply follows that if they can't explain it conceptually then they must not understand it conceptually. And that simply means that they don’t really know what they are doing. It's not a personal judgment. It's a logical conclusion pure and simple.

Further, if geometry is a quantitative thing, ie coordinates and lengths, then if I have an idealized right angle triangle of sides 1,1 and sqrt(2), and sqrt(2) is irrational, then how come irrational numbers don't represent quantity?

Which came first? Geometry or the concept of number? I hold that the concept of number came first, and that had it been defined "properly" (as a correct model of the quantitative nature of our universe), then irrational numbers cannot be said to exist by that definition. (it's simply a matter of definition and the fact that irrational numbers do satisfy that definition therefore we can't think of them as valid numbers in terms of being individual things). You guys love axioms. Well, this is basically doing nothing more than taking the "corrected" definition of number and treating it like an axiom. Any number that doesn't satisfy this definition (or axiom) cannot be called a number simply because it doesn't fit the definition or satisfy the axiom. So while we can play with these irrational "concepts" we can't officially recognize them as numbers because they don't formally fit the "corrected" definition of number.

That doesn’t mean that we can't still use them as concepts other than numbers. In fact, they have much value in that way. We certainly couldn't get by without the concept of irrationality. But we would do well to also recognize why they don't satisfy the definition of number. That in and of itself is enlightening.

I'm not sure what "unit" has to do with anything.

Well, if the number One is defined on a concept of "unity" then to suggest that you have a "unit" of something is to refer to the number One.

In other words, the whole concept of geometry rests on the concept of numebr there's just no getting around it.

And it got along fine for a very long time without coordinates...

I would disagree with this in the sense that it really doesn't even make any sense to talk about something like a finite length without implying the concept of coordinates.

I realize that formal coordinate systems weren't fully defined at the time of Euclid, but that doesn't mean that he wasn't using the concept. In fact, to even talk about the concept of two points,… (there's the number 2 right there, so for Euclid to talk about more than 1 point he was necessarily using the concept of number already). In any case, to even talk about the concept of two points which are not the same point automatically infers the concept of a crude coordinate system. Euclid just didn't acknowledge his use of this basic intuitive idea.

Furthermore, coordinates have attained a reputation for obscuring geometrical facts and ideas (and I agree). They're good for computation, but not for understanding.

I'm not sure that I agree with this at all. Like I said above, if your geometry refers to more than 1 point you are automatically implying a concept coordinates whether you are aware of it or not. In other words you can't even talk about a concept of length with implying that you have at least two points involved that are not the same point.

Maybe that's because you aren't familiar with some of the problems with the alternatives.

You mentioned that you have a problem with irrational numbers. Well, there is an affine plane whose coordinates are rational numbers. But it suffers from the problem that lines can pass through the center of a circle without intersecting the circle.

I'm not even going to attempt to respond to this in a short post, but given the correct forum and time I would not shy away from addressing it. A topic like this deserves an in-person discussion and a pot of coffee.

Another problem shows up with functions. I'm sure you're familiar with the intermediate value theorem:

Thm: If a < b, f is continuous on [a, b], f(a) > 0, and f(b) < 0, then there exists an c in (a, b) such that f(c) = 0.

The thing is, when you don't live in a continuum, then the intermediate value theorem can fail.

Yes, I'm familiar with the Intermediate Value Function. I see no problem here at all. To say that a function is continuous does not mean that it represent a continuum. Where did you ever get that idea? It certainly doesn’t follow from the mathematical definition of "continuous". I forget the precise definition off hand, but for a function to be continuous as we need to do is prove something about it's left-hand and right-hand limits (like they represent the same quantity "have the same value or are equal"). That doesn’t prove that a function is a continuum. UNLESS, you believe that a finite line segment is necessarily a continuum. But recall, what with the "corrected" definition of number it is conceptually understood that a finite line segment cannot be continuous (i.e. it cannot contain an infinite number of points).

So this is all quite compatible with a corrected definition of number and there are no problems in all of calculus I've already checked it out. There is nothing in all of calculus that would be affected by this corrected definition of number. Actually I wish there had been because I could have used that as additional prove that a change needs to be made. Unfortunately I need to get into Group Theory for that and I have absolutely no education in group theory at all so I can't quickly see where the problems will be. I wish I could, because finding a problem would give me even more evidence that this change must be made. Either that or it will prove to me that I am wrong. But I seriously doubt that. That's not an egotistical statement by the way. It's just a conclusion based on logic.

In any case, there is nothing in calculus that conflicts with changing the definition of number as I am proposing. However, there may be differences in interpretations. In fact there definitely WILL be differences in interpretations. Things will make more sense.

NOTE TO EVERYONE

Thank you all for the conversation. I really don't have time to do this anymore and it seriously isn't my intention to convince any of you of anything. I am simply getting hung up here on responding to concerns.

I didn't come here to sell my ideas to anyone. My original purpose was to simply point out that current axiomatic mathematics is deficient. By simply defining number conceptually correctly addition could be proven. That's really all I wanted to say.

In some ways I'm sorry that I even brough it up.

I just don't have time to try to explain this to a hostile audience.

I really have better things to do with my time. I do appreaciate the interest in my views and intelligent style of discussion. I wish you all the best. But I really need to quit wasting my time here. I have other things that need attention.

Again, thank you all for your interest.

 

[matt grime]

Two sets have the same cardinality if there is a bijection between them. A simple concept. If the sets are finite this is iff they have the same number of elements.


Modulo some very annoying set theory we can declare the cardinal numbers to be equivalence classes of sets modulo bijective correspondence. (annoying because we need to avoid russell's paradox).

The cardinal numbers are partially ordered. If we assume the axiom of choice then it is even well ordered, and that is the system Cantor developed of the alephs. The class of cardinals is not a set in ZF, or even ZFC.

At no point have I needed use the notion of quantity. Quantity as an interpretation comes afterwards, and often obscures the real point.

I noticed you didn't respond to Hurkyl's comment that geometry is consistent (the tarski comment) and doesn't even have the power to define integers (ie quantities).

One can use cardinals to show, eg, that transcendental numbers exist. Indeed, the algebraic numbers have measure zero in the reals.

 

[Hurkyl]

Bijection? Can you explain what that means conceptually.

I can't tell if you're asking because you don't know, or just want to task us to explain it. A bijection is an invertible function.

It's a relation between two classes of "things" such that each object from either class is related to exactly one object from the other class.


"Counting" is the most basic example: the act of counting is simply the process of deciding upon a bijection. For example, when I'm counting the three pennies on my desk, I pick a penny and call it "one", I pick another and call it "two", and pick the last and call it "three". I've formed a bijection between {one, two, three} and the pennies on my desk.


Another example is the proof that any two line segments have the same cardinality. The core of the proof is as follows:

Let AB and CD be line segments, and let the lines AC and BD intersect at some point E. Define a function f from AB to CD as follows:

Let X be any point on AB. By the crossbar theorem, the line XE intersects CD at some point, Y. Define f(X) := Y.

It is fairly straightforward to see that this is a bijection. Thus, the segments AB and CD have the same cardinality.

(A clearer way to define f is to be the set of points (X, Y) in ABxCD such that X, Y, and E are colinear.)

had it been defined "properly"

"Properly" means that "number means positive rational number"?

In other words you can't even talk about a concept of length with implying that you have at least two points involved that are not the same point.

Who needs to talk about quantity? One of the Hilbert axioms of Euclidean geometry is:

There exists points X, Y, and Z such that X != Y, Y != Z, X != Z, and that Z does not lie on the line through X and Y.

While it defines the existence of three distinct noncolinear points, it doesn't invoke any notion of quantity.


Just because the notion of quantity can be used to describe something doesn't mean that you can't speak about that something without referring to quantity.

 

[NeutronStar]

Just because the notion of quantity can be used to describe something doesn't mean that you can't speak about that something without referring to quantity.

I buy that, and I agree that your specific example with geometry was a non-quantitative idea. But I would also argue that that particuar example also has absolutely nothing at all to do with any idea from set theory.

I never meant to imply that other forms of logic are invalid. I use non-quantiative forms of logic all the time. Hell, I was a computer programmer using machine language. I've used Boolean Algebra a lot. That has absolutely nothing to do with ideas of quantity. But again, it doesn't rely on set theory either.

I'm obviously being grossly misunderstood here if you believe that I think that every imaginabe concept can be reduced to an idea of quantity. I've never said or implied any such thing.

Now I have said that any meaningful idea of "number" can be reduced to an idea of quanity. But that's no where near the same thing as claiming that every conceivable concept can be reduced to an idea of quantity. I would be quick to reject such an absurd notion myself.

I can't tell if you're asking because you don't know, or just want to task us to explain it. A bijection is an invertible function.

It's a relation between two classes of "things" such that each object from either class is related to exactly one object from the other class.

Yes, I'm fully aware of what bijection is.

I'm also fully aware of Cantor's diagonal proof that the real numbers cannot be put into a bijection with the natural numbers. I agree with his proof, but disagree with his conclusion.

There is a qualitative difference between N and R that shows up in bijection. But there is no quantitative difference between |N| and |R| as Cantor implies by claiming that they have different cardinalities. Of course, he's kind of stuck with his conclusion because of how he had defined the number One as the set containing the empty set. I mean, based on Cantor's set theory he has no choice but to come to the conclusions that he has come to.

In a "corrected" set theory the set of real numbers wouldn't be a valid concept to begin with. We could still talk about the real numbers, we simply wouldn't be able to talk about the SET of real numbers. Because irrational quantities don't qualify as "numbers" by definition. We could still talk about the set of "Solutions" to equations and list irrational numbers in that set. But that's a whole different story. That wouldn't be a set of numbers, it would be a set of solutions.

In a very real sense Cantor is wrong. The difference between |N|, |Q| and |R| is not a quantitative difference. It's a qualitative difference. Yet it is a qualitative difference that |N| and |Q| do not share. In other words, qualitatively speaking N, Q and R are all distinctly different from each other. I hold that quantitatively |N|=|Q|=|R|. In other words there is no quantitative difference between them. Yet at the same time I can clearly see that |N| and |Q| share a common quality that |R| does not. And THIS is why they can't be put into a bisection.

So there are three things going on here.

1. There is the fact that N, Q, and R all have different qualities. (everyone already knew that)
2. There is the fact that |N| and |Q| share a common non-quantitative quality that |R| does not.
3. And then there is the fact that quantitatively |N|=|Q|=|R| (which the mathematical community would disagree with)

The mathematical community is unaware of this.

They think that quantitatively |N| = |Q| != |R| . Except they don't like to think of it solely in terms of quantity because this leads to a paradox. So they just call it "cardinality" instead of "quantity" and leave it at that.

In other words they have no clue what the quality is in statement number 2 in the above list.

This is at least one of the conceptual differences between current set theory, and a corrected model.

 

[Hurkyl]

I mentioned at one point I thought your argument was more about semantics than anything substantive. Some of your comments in this latest post support this characterization:

Because irrational quantities don't qualify as "numbers" by definition. We could still talk about the set of "Solutions" to equations and list irrational numbers in that set. But that's a whole different story. That wouldn't be a set of numbers, it would be a set of solutions.

Whether you call them "algebraic numbers" or "solutions to polynomials", it's still the same thing: the only change is the name you've picked.

I hold that quantitatively |N|=|Q|=|R|.

And here, you, for some reason, don't like the fact that the notation |.| means cardinality. So you've decided to reassign |.| to refer instead to your vague notion of quantity.


How did you determine that the "quantity" of elements in N, Q, and R are all the same, anyways? I get the impression that, one day, you decreed that "quantity" means either a positive integer or "infinity". From this you rejected the empty set because you couldn't apply your notion of quantity to it, and you decided N, Q, and R are all have the same quantity because, well, they aren't finite so the only thing left is "infinity".


Frankly, if that's all your notion of quantity is, it's not particularly useful. Mathematicians already know about positive integers and infinite sets.

(I still don't yet know if you accept or reject positive rational numbers)


Anyways, Matt Grime hinted at this, but there are other serious issues to consider than set theory. For example, if you're considering the rational line, there's a serious problem with the concept of "length"

For example, we would like the "length" of the interval [a, b] to be b - a, right? Well, the problem goes like this:

The set of rational numbers are countable: there is a bijection, f:N->Q

So, we can set about "covering" the entire rational line with these intervals:
[f(1) - 1/2, f(1) + 1/2]
[f(2) - 1/4, f(2) + 1/4]
[f(3) - 1/8, f(3) + 1/8]
[f(4) - 1/16, f(4) + 1/16]
...

The lengths of these intervals would be 1, 1/2, 1/4, 1/8, ... respectively, and they have a total length of 2. Yet, they cover the entire rational line! (because f(n) is in the n-th interval, and f(n) enumerates all the rationals)


We have the problem here that a bunch of intervals can combine to form a new interval whose length is greater than the sum of the lengths of the individual intervals!


However, length as defined for the real line doesn't suffer from this problem.



And one more comment: the term "number" is only used for historical reasons.

 

[Hurkyl]

Oh, and on Cantor's natural numbers.

(side comment: just like Camelot in Monty Python, "it's only a model")


Cantor did not represent 1 as {{}} because he thought it would be nifty to represent 1 as the set containing the empty set. He chose to represent natural numbers as the set of (representations of) smaller natural numbers.

There are no natural numbers smaller than 0, so 0 is represented by {}. 0 is the only natural number smaller than 1, so 1 is represented by {representation of 0}, which is {{}}.

 

[NeutronStar]

Whether you call them "algebraic numbers" or "solutions to polynomials", it's still the same thing: the only change is the name you've picked.

I disagree. It's not just a matter of semantics. I'm saying that the list of solutions of any polynomials that happen to contain irrational numbers wouldn't satisfy the defintion of a mathematical set, so you can call them anything you want as long as you don't call them a "set".

I suppose in a sense that is semantics. But if someone gave you a relationship that was not a function and they told you it was a function you would argue that it isn't because it doesn't satisfy the definition of a function.

Is that "just" sematics?

That's all I'm saying here. Irrational numbers don't have a property of individuality and therefore cannot be considered to be a element in a set (in my formalism). Therefore, by definition, it would simply be incorrect to claim that such a collection qualifies as a mathematical set. It simply doesn't have the correct properties and fails to satisfy the defintion of a set (in my formalism). Just like some relationships don't qualify as functions. It's no different.

In fact, just as mathematics recognizes relationships that aren't functions. I can also recognize other types of collections that aren't "sets" by definition. I don't have to toss out the concept of irrational objects altogether. I simply need to treat them differently than numbers.

It's a matter of satisfying the definitions of the formalism. That's all.

My formalism recognizes that irrational concepts are not the same thing as the concept of number. They have a different property. This unique property deserves to be treated in a different conceptual way. After all, it ultimately is a different concept. Why not deal with it properly? Why just ignore it and lump it in with the concept of number which is a different concept altogether?

Actually I don't know why I keep responding to this thread. I really don't care whether you understand what I'm saying or not. Honest.

I can only say that no one has said anything at all that has caused me to question my position in the least.

So, we can set about "covering" the entire rational line with these intervals:
[f(1) - 1/2, f(1) + 1/2]
[f(2) - 1/4, f(2) + 1/4]
[f(3) - 1/8, f(3) + 1/8]
[f(4) - 1/16, f(4) + 1/16]
...

The lengths of these intervals would be 1, 1/2, 1/4, 1/8, ... respectively, and they have a total length of 2. Yet, they cover the entire rational line! (because f(n) is in the n-th interval, and f(n) enumerates all the rationals)

Ok, I don't have time to think about this right now. But it does look interesting. I WILL look at it in detail, because it does look interesting, but I may not have time to do this for several days. Going through this little problem is bound to be insightful one way or the other so it will be worth doing. Sometimes it takes a while to figure out why something that seems obvious is actually incorrect.

You don't happen to have a URL on the Internet that explains this problem. Just glancing over it here I'm not sure I fully understand precisely what you are doing. This must be a popular proof, it should be on the Internet somewhere. Does it have a name?

 

[zefram_c]

Here is the proof that a bijection f(x) can be found from N to Q: http://planetmath.org/encyclopedia/ProofThatTheRationalsAreCountable.html

Now we have f(x):N -> Q. So Q = {f(x): x is in N}.

Then Q is contained in the set { [f(x)-a, f(x)+a]: a=2^-x, x is in N }

This gives you a covering of Q of total length SUM (x = 1 to inf) {2^-x}=1