Prove Addition

[Hurkyl]

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


Things have changed slightly since Pythagoras's time. :tongue: 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. Ive noticed that you realised 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 refering 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 [Broken]

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 quanity 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. :yuck:

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 dont need to worry about addition being provable, becuase you use it in the tense that we define it as.

I hope this helps. :surprised

 

[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 possess 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?