What makes mathematics?

  • #1
On Radioactive Waves
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We use theorems and postulates based on axioms. We use set theory. Are axioms based on set theory? or vica versa? More specifly, what makes mathematics? What is it's primaray basis?

It seems to me the entire basis of mathematics is the number 1, and its relation to itself.


I know one of you math wizzes should be able to explain this...
 
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Answers and Replies

  • #2
selfAdjoint
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Sets are more basic than 1 - you can get the numbers out of sets with for example the Zermelo-Frankel (ZF) axioma.

These axioms (and competing ones) show how you can combine sets to develop almost all of mathematics. It is these axioms that are subject to the Goedel incompleteness theorem.

Working mathematicians usually don't bother with the ZF proofs of their areas, though they acknowledge that they exist. Usually the ZF proofs are at too low a level to be useful, like machine code versus a programming language.
 
  • #3
STAii
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Can you tell more about the "Zermelo-Frankel (ZF) axiom" ?
Thanks.
 
  • #4
selfAdjoint
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Well here is a link, http://www.trinity.edu/cbrown/topics_in_logic/sets/node4.html , but I don't think it will help you much. The upside down A means "for all", the uside down E means "There exists". Letters like x or y are elements of sets.
 
  • #5
NeutronStar
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Originally posted by On Radioactive Waves
We use theorems and postulates based on axioms. We use set theory. Are axioms based on set theory? or vica versa? More specifly, what makes mathematics? What is it's primaray basis?

It seems to me the entire basis of mathematics is the number 1, and its relation to itself.

First off let me explain that I am not a mathematician. If I were I would quickly be kicked out of the community.

However, I believe that I can safely say that the ZF axioms are far too advanced for the question you are asking. I say this because the ZF axioms already make use fundamental axioms that seem to be more in line with what you are asking about. The ZF axioms are basically useless to me because I have a problem with mathematical formalism at a far more fundamental level that that.

Even without being a mathematician I am certain that I can safely say that set theory is based on axioms (not the other way around). I can also safely say that mathematics is primarily based on set theory. And while this may seem absurd, set theory is based on nothing. Yes, this is an absolute truth.

Just over 100 years ago the mathematical community was pressured into formalizing the definition of the natural numbers. There were many key players in this development. Gottlob Frege, Giuseppi Peano, Leopold Kronecker, Henri Poincare, and of course, Georg Cantor to name only a few. Needless to say Georg Cantor came out the winner with his empty set theory based on nothing.

Henri Poincare had this to say about Cantor's set theory, "Cantor's set theory will be considered by future generations to be a disease from which they must recover".

I totally agree with Henri Poincare on his point.

Cantor's set theory (which is the basis of all of modern mathematics) has as its foundation the idea of nothing disguised as an empty set. This empty idea is used purely subjectively to define a qualitative idea of the number One. All of the other Natural Numbers (and every conceivable mathematical idea of number) rest on this qualitative subjective idea of nothing.

The mathematical community over-all was pleased with this result. The reason being is that they wanted to develop a 'pure' idea of quantity which was not associated with any 'thing' that is being quantified. Cantor's qualitative approach to subjectively quantify the idea of nothing was very attractive for this goal.

I believe it is very important to realize that the mathematical community at this point in time was much more concerned with the idea of purity in thought, than they were with preserving any real definition of quantity. It was at this very moment that mathematics departed from out intuitive understanding of the quantification of 'things' and moved into a completely abstract (and vague) formalism of a subjective qualitative nature.

For me personally, it is totally absurd to believe that we can invent an idea of quantity apart from the idea of the things that are being quantified. To do such a thing is to move away from our intuitive understanding of the concept, and invent a completely meaningless abstract model that has nothing at all to do with our original intuitive idea of quantity. How the mathematical community could make such an obvious wrong turn is completely beyond my comprehension to understands. I'm sure it was completely beyond Henri Poincare's comprehension as well. I might add that Kronecker, and Peano were not pleased with Cantor's idea either.

So today the Number One is officially defined as the property of a set containing an empty set. This is Cantor's genius. One problem with this, and there are many, is that the idea of an empty set has bestowed upon the idea of a set a property that goes beyond the intuitive idea of a collection of things. I mean, the very idea of an empty set is an idea of a collection of things which is not the collection of a thing. Seems like a pretty obvious logical contradiction to our intuitive idea of a set as a collection of things.

Not so, the mathematicians say. We have axioms!

The problem is that their axioms are inconsistent. Gödel has proven this beyond a shadow of a doubt. Mathematicians have been trying to patch up Cantor's empty set theory ever since they so boldly accepted it.

Here's one such patch: There is a distinction between an element and a set containing an element.

Don’t ask a mathematician to explain this distinction, they can't. They will simply reply, "It's an axiom", or an appendum to an axiom, or a band aid, or something like that. I'm not even sure if all mathematicians will agree.

However, it should be quite clear that the distinction must be made. Zero is defined as the property of empty set. One is defined as the set containing an empty set. If there is no distinction between an element, and a set containing an element, then it follows that there is no difference between the numbers Zero and One! Ouch!

The result of this distinction is that a set must have some phantom property beyond the property given by the collection of things that make up the set. But no one knows what this phantom property is. They just try to shove it under the carpet and hope that it doesn't make too big of a lump.

In truth, this phantom property is bulging out all over the place causing all sorts of problems, particularly when the mathematics get into the realm of the very abstract such as in the mathematics of quantum mechanics.

Georg Cantor was the only human in history to start out with nothing and end up with more than everything (infinities larger than infinity). This absurd concept is a direct result of his subjective qualitative approach to defining the idea of quantity based on an idea of nothing. It's not merely abstract, it's totally absurd. It's also inconsistent (just ask Gödel), and it's even a logical contradiction - A collection of things that is not a collection of a thing? Please! How the mathematical community could ever accept that one is beyond me.

Well, I'm sure that this post will likely stir up some responses. I hold that everything that I've said here is true. More importantly I hold that I have a solution to the whole problem. Well, actually it is Giuseppi Peano's original solution. In my opinion the mathematical community should have listened to him. Mathematics would be in much better shape had we taken that path. Not to mention that Gödel's proof would no longer apply to mathematics. Not that Gödel was wrong, he wasn't, it's just that if we had followed Peano's lead in defining the Natural Numbers Gödel's inconsistency proof would simply no long be applicable to mathematics. It would be a completely different formalism, albeit it wouldn't look much different to engineers who use calculus. We need to keep in mind that calculus was invented prior to Georg Cantor and set theory as based on the empty set. So changing set theory won't have much of an effect on calculus, or just about any mathematical concept that was introduced prior to 100 years ago.

That's my 2 cents. :wink: And keep in mind that I'm not a mathematician. No mathematician will agree with me. They can't. It would lose them their job!

By the way, here's a link to a really great mathematics site that has historical information on just about every mathematician who ever lived.

http://turnbull.mcs.st-and.ac.uk/history/
 
  • #6
Hurkyl
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The point to the axiomatic method is to clarify the properties held by the objects in question. They are like a user's manual; they give a list of all the basic things you can do with an object.

The axiomatic method is not a recent invention; Euclid used it in The Elements. The only thing new about it is we understand it better now, and we use it far more pervasively.


The benefits of the axiomatic method are not just the precision and clarity it allows; the clarity makes connections between seemingly unrelated ideas possible, and it permits "theorem reuse"! For example, a lot of the properties of the real numbers depend only on its arithmetic and don't care at all about their structure (i.e. the fact they're ordered and connected). One practical application of this is that the vast majority of the theory of solving linear systems of equations is directly applicable to other nice number systems (such as arithmetic modulo a prime).



On another matter, one of the most important discoveries in mathematics was the notion of relative consistency. The "parallel postulate" was a thorn in the side of geometers for millenia; the other axioms seemed quite self evident, but even back in Euclid's time people realized that the parallel postulate wasn't quite so obvious, so they thought they should be able to prove it in terms of Euclid's other postulates. One approach to trying to achieve this goal was to assume the parallel postulate was false and try to derive a contardiction, thus proving it had to be true. A few mathematicians explored this geometry in exquisite detail, developing what we now call hyperbolic geometry.

Eventually, it was discovered that hyperbolic geometry and Euclidean geometry were relatively consistent, meaning that if you could find a logical contradiction in one of these theories, then you could find a logical contradiction in the other. (incidentally, Poincaré was one of the people who contributed greatly to this realization) Mathematicians constructed a model of hyperbolic geometry in the Euclidean plane; a system of things one could call "hyperbolic lines" "hyperbolic points", et cetera. Using the axioms of Euclidean geometry, you can prove that this system of hyperbolic things satisfies all of the axioms of Euclidean geometry except the parallel postulate, which it violated.

The consequence of this is astounding; this meant that if geometers ever succeed in proving the parallel postulate from Euclid's other axioms, they could also prove the parallel postulate in this hyperbolic system. However, Euclidean geometry was able to prove that the parallel postulate was false in the hyperbolic system, so that would mean that there was a contradiction in Euclidean geometry!

That means, If geometers could prove the parallel postulate, they would thus prove Euclidean geometry was self-contradictory! To state it another way, if Euclidean geometry is self-consistent, then Hyperbolic geometry is self-consistent!

(the converse of this statement is true as well, but that requires an additional proof; a model of Euclidean geometry built in hyperbolic geometry)



The idea of relative consistency turns out to be a useful one. Godel proved that if any (sufficiently powerful) theory that could deduce its own self-consistency must be inconsistent. Thus, if we want to build confidence in the consistency of the complicated theories we like to use, we have to turn to a different theory to prove it... generally a simpler theory that is "more obviously correct". The idea is the same as what happened with geometry; we build a model of the complicated theory using the axioms of the simple theory.


Which leads us to the formalism of mathematics in terms of set theory. We have to remember that the point of set theory was originally not to provide a foundation for all of mathematics; it's point was to provide axioms for working with sets! It gives us, for example, the notion of set unions, set intersections, and power sets... things all "more obvious" than the complicated structure of, say, the real numbers.


But because these operations are so obvious, it is a great confidence builder to be able to model complicated things (like the real numbers) in terms of simpler theories like set theory and natural numbers. Incidentally, one of the axioms of ZF (the axiom of infinity) says essentially "There is a set of all natural numbers"... so we model everything using ZF as the foundation because it includes both the axioms for sets and for natural numbers. (though the ZF axioms are very minimalistic, so it's not obvious that this is what they actually say until you do some work)

By doing this, we gain greater confidence in the self-consistency of our more complicated theories because we can model our complicated theories in terms of set theory which is comparatively obvious.


Now that that's explained, some particular responses:

And while this may seem absurd, set theory is based on nothing.

You can prove there is an empty set from the axioms of set theory, and that sets built up from the empty set are sufficient to model any mathematical theory... but the axioms don't forbid the existence of other sets. Things build out of the empty set are just a model. (I'll leave it to another topic to discuss the ramifications of models) Building things out of the empty set is a minimalist thing, because every time we say "Let's take as an axiom the existence of the set S", we have to worry whether that axiom is consistent with everything else... so for the goal of proving relative consistency, we model everything using just the empty set.

But by no means do we suppose that is the only model of our theory! In fact, the great thing about the axiomatic method is that it allows is to completely ignore the messy details about precisely what the objects in question are... all that matters is how we work with them. We typically have some imaginative picture in our head about what they are, but if we're careful and make sure every step in our logic can be rigorously justified from the axioms, we avoid having any problems with our mental picture leading us astray.


To do such a thing is to move away from our intuitive understanding of the concept, and invent a completely meaningless abstract model that has nothing at all to do with our original intuitive idea of quantity.

On the contrary; the point to axioms is to distill the essense of our intuitive "understanding" into precise rules to follow. (and then we model these rules with set theory, or something derived from it, to make sure that we haven't gone awry and have written down self-contradictory rules)

Typically, the progression of a new concept starts off with an ugly expression of the concept in terms of concepts we all ready know, then as more knowledge is gained about the concept we can finally cast it into a clear axiomatic formalism (aka abstraction) that allows us to cast away the messy details and work just with the concept we're interested in. It would be nice to have this clear axiomization right from the beginning, but it's usually not clear how to do it immediately; it's far easier to define the rules of manipulation in terms of objects we already understand instead of trying to define rules of manipulation from scratch!


So today the Number One is officially defined as the property of a set containing an empty set.

Not quite; today, the number one is typically defined as the multiplicative identity. (at least in systems where we wish to have an element we call "1"... for example, with matrices, we usually call the multiplicative identity "I")


Here's one such patch: There is a distinction between an element and a set containing an element.

Don’t ask a mathematician to explain this distinction, they can't.

I can eat cookies. I can't eat ideas. A set is an idea; I can't eat a set of cookies, but I can eat the cookies in a set. That's the intuitive meaning. (Of course, we typically don't speak so precisely, and we would usually say "I ate a set of cookies" instead of "I ate the cookies in a set of cookies" because it's shorter to say)


Not that Gödel was wrong, he wasn't, it's just that if we had followed Peano's lead in defining the Natural Numbers Gödel's inconsistency proof would simply no long be applicable to mathematics.

We do use Peano's axioms as the definition of the natural numbers. And Godel's theorems, IIRC, do not use set theory; just logic and number theory (aka the theory of the natural numbers, +, and *).



Incidentally, I have read that a few people think that category theory could suffice as an alternative to set theory for the foundation of mathematics.
 
  • #7
NeutronStar
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Incidentally, I have read that a few people think that category theory could suffice as an alternative to set theory for the foundation of mathematics.

This doesn't surprise me in the least since I see Cantor's set theory as exactly that. It is based on category (or quality) rather than an idea of quantity. If that isn't categorization I don't know what is. The whole purpose of Cantor's empty set is to separate the idea of quantity from the idea of the things that are being quantified. As far as I'm concerned, once that has been accomplished our intuitive idea of quantity has already been abandoned. Anything goes after that. It's a free for all.

So yes, our current understanding of set theory could easily be expressed as a category theory because this is basically all that it is to begin with.


We do use Peano's axioms as the definition of the natural numbers.

Yes, I agree with this. But I must hasten to add that the Peano axioms that we use today are the watered-down version of his original proposal. In other words, Peano was there when Cantor presented his empty set theory. Peano had no choice but to accept what the rest of the mathematical community accepted, and work within the confines of Cantor's definition for the number One. Instead of defining the number One Peano merely used it as a 'given'. His first axiom has been reduced to nothing more than the statement "1 is a natural number". He then goes on from there. He personal definition of the number One was rejected.

His personal definition for the number One, by the way, had to do with the idea of unity. The mathematical community wanted to know what he meant by 'unity'. He could no give them a working definition without referring directly to some 'thing' that is being quantified. For this reason he was not permitted to use his idea of unity. So he simply started with the axiom that the number 1 exists without definition. It was a setback to be sure, because there is much to be gained by including the idea of the thing that is being quantified when working with the idea of quantity.

So while I can't disagree with your words, I do disagree with the spirit. We actually don't use Peano's original ideas.


You can prove there is an empty set from the axioms of set theory,…

I will have to beg to differ with you on this point. To begin with if you can prove it then why do we need the axiom "There exists an empty set" [?]

I mean, if it's provable, then why do we need an axiom for it [?]

I actually have a mathematics textbook around here somewhere that directly states that the empty set is not subject to proof. Here is it,… Foundations of Higher Mathematics ISBN 0-87150-164-3. (By the way On Radioactive Waves, you might enjoy looking at this book for a very nice elementary introduction on Cantor's set theory. Although, the bulk of the book is actually an introduction to conjecture and proof). I wouldn’t recommend buying it, but if you can find it at a library it might be interesting to look through.

Hurkyl

I will basically agree with you that mathematics is a somewhat sound, stand-alone, purely abstract valid axiomatic formalism. I never did doubt that. My concern is that it doesn't correctly reflect the true nature of the property of our universe that we call quantity. And this is my only concern.

I do not wish to attack mathematics as a pure abstract formalism. However, if mathematicians wish to claim that mathematics correctly represents the physical properties of our universe that we call quantity then it had better convey them correctly. But I don't see how it possibly can now that the idea of quantity has been formally divorced from the idea of the things that are being quantified.

I am thoroughly convinced that our current abstract mathematics does not correctly represent physical reality. Obviously, engineers tend to be careful to use units of measure in everything that they quantify (we call that Applied Mathematics and we pretend that it is just a sub-set of Pure Mathematics when in fact it is not). It is a completely different animal. We totally ignore the empty set when we use Applied Mathematics, and since we seldom (if ever) work with infinite quantities we don't run into any problems.

However, when we start working with lofty ideas like quantum mechanics we must rely on the totally abstract methods of pure mathematics. Only then do we start to get into trouble.

On Radioactive Waves

My intention on this thread is to genuinely share with you some of my concerns about the formalism of modern mathematics. I firmly believe that all of my concerns about mathematical formalism have merit. If you are just now beginning to look into mathematical formalism at the level of set theory be aware that everyone does not agree with the idea of an empty set (many famous mathematicians who were alive at the time the idea was accepted did not like it at all)

I personally feel that they had many sound reasons to object to the idea. If I were alive at the time I too would have vehemently objected. I believe that Cantor's level of abstraction was not merely a move toward abstraction, but it was a move toward the absurd. It is not logical, it does not represent the intuitive nature of physical quantities.

I am not saying to simply ignore mathematics. On the contrary, dig into it as deeply as you can. Most of mathematics is absolutely beautiful and powerful. Cantor's set theory is the ultimate of human ignorance. However, set theory done correctly is one of the most beautiful concept of the human imagination. So by all means, study it to the hilt. But if you find yourself uncomfortable with any of Cantor's ideas, please don't be afraid to question them to their core. I believe that if you do you will eventually see the error in their logic.

Euclid's Geometry:

By the way, Euclid's geometry can indeed be understood outside of a purely axiomatic approach. In other words, just because he used an axiomatic method to arrange his ideas doesn't mean that this is the only way his ideas can be comprehended.

As an example. Euclid begins with a point as an axiomatic elementary entity in his geometry. But it is just as possible to being with a point as a location. (a concept, not a mere axiom).

I realize that everyone's next question would then be, "Please explain what you mean by location. Well, it's not my intent to go through the whole explanation here, but I will say this much. If we are talking about a single point then the idea of location is meaningless. I would still say that a point is a location however, and it would all become clear as we continue to build the formalism by adding more points. We would then clearly see and understand that location is a relative concept and cannot be understood within the concept of a single point. Does this mean that we can't have a single point? Absolutely. What value does a geometry have that contains nothing but a single point? It's meaningless.

So now we find ourselves talking about two points? What does location mean. Well, if the two points that we have are not the same point, then they necessarily must have different locations, etc, etc, etc. We can build up an entire geometry from there. No axioms necessary, all we need are genuine comprehensible concepts. We can set Euclid's geometry free from it's fundamental axioms. My kind of formalism. :wink:

Try this with Cantor's empty set. It won't work. Why? Because the idea of an empty set cannot be comprehended in and of itself as an entity. Unless, of course, we wish to tackle the problem of defining that phantom property that it simply must possess. Thus the need for an axiom (to avoid having to define it). Cantor's set theory cannot be comprehended without the crutch of axioms. It is forever bound to them never to be set free.
 
  • #8
Hurkyl
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If I care about the particular identities of the objects under consideration, I work in a system that respects their identities... but in general I don't need to burden myself with a complete description of the objects in question.

This isn't just a math phenomenon; for example, at the store, I might by a box of assorted donuts, or a can of mixed nuts. If I care about the details, I can inspect the donuts to see they are plain, sugar powered, and cinnamon powdered, but I don't have to.


The axiomatic formalism permits you to consider only the properties important to the issue at hand, but it does not force you to do so.



... no axioms necessary, all we need are genuine comprehensible concepts...

But your system has an obvious problem; my notion of location may or may not be the same as your notion of location. For example, my notion of location would certainly permit the universe to have only one location (but empircal observation proves otherwise). The value of an axiomatic formulation is clear here; while our ideas about the fundamentals might differ, as long as we can agree that these fundamentals satisfy the axioms of, say, Euclidean geometry, we then have a common foundation upon which we can share ideas.


Because the idea of an empty set cannot be comprehended in and of itself as an entity.

To be honest, I never understood why people have trouble with concepts like "zero" or "empty set"; I've always found them to be clear, practical concepts... in other words, I'm certainly not going to accept this statement as a given.


And as I mentioned, ZF does not need an axiom to prove there is an empty set; all ZF needs is to know that there exists at least one object which we can discuss. The concept behind the proof is very clear and can be demonstrated with an example.

Harry, Meg, and Joseph empty their pockets and place a bunch of coins on the table. Which of the coins on the table belong to George? The answer to this question is the empty set.
 
  • #9
On Radioactive Waves
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Wow, !

Thanks Hurkyl (whose appearance doesn't surprise me) , and Neutron Star, very nice responses. I wasn't sure if I stated my question right, but this was exactly what I was looking for.


I will need to explore the information here, there's enough to keep me busy for a while ( so stay out Lethe and Hallsofivy and there
s a couple more people I wouldn't mind hearing from)
 
  • #10
lethe
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Originally posted by On Radioactive Waves

I will need to explore the information here, there's enough to keep me busy for a while ( so stay out Lethe and Hallsofivy and there
s a couple more people I wouldn't mind hearing from)

so you don t want a reply from me?

well, i think hurkyl has this one pretty much in hand, but i guess i want to go down on record as saying that neutron star is way off base, and misses the point entirely of abstraction in mathematics, so take what he says with a grain of salt, that s my opinion, at least.
 
  • #11
On Radioactive Waves
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No, I always learn things from you Lethe. I was just saying I had enough to look at already for now... but if you disagree with what's been stated by all means do jump in- I'm learnig just watching this discussion.
 
  • #12
NeutronStar
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hurkyl wrote

If I care about the particular identities of the objects under consideration, I work in a system that respects their identities... but in general I don't need to burden myself with a complete description of the objects in question.

It doesn't matter how you view your objects if you are working with modern mathematics, the rules are the same. You are free to treat infinite objects as though they have a quantitative property of One. That is totally absurd. Yet it is the basis of Cantor's set theory. Cantor's set theory not only permits infinite objects to be counted as having a quantitative property of One, but it is based on this very concept - the empty set itself.

hurkyl wrote

The axiomatic formalism permits you to consider only the properties important to the issue at hand, but it does not force you to do so.

I would certainly hope that mathematics would demand that the property that one should focus on is the property of quantity . Unfortunately Cantor's set theory permits focusing on quantitative properties instead. Not only that, but it doesn't even permit the recognition of quantitative properties actually.

Cantor's set theory is absolutely positively a category theory. It has nothing at all to do with the idea of quantity.

hurkyl wrote

To be honest, I never understood why people have trouble with concepts like "zero" or "empty set"; I've always found them to be clear, practical concepts... in other words, I'm certainly not going to accept this statement as a given.

This only goes to show that you have accepted mathematics for the category theory that it is. You have no problem with it because you aren't trying to justify it with your comprehension of the idea of quantity, or with what you intuitively understand as a quantity of One.

Either that, or you have simply accepted the idea of a set as being a container rather than as the collection of things that make up the set. If this is true than you have really missed the quantitative concept of number altogether.

I have no problem at all with the concept of "zero". Zero is the absence of quantity, and therefore it is the absence of a set. Because of this zero is not a number. It can't be, because the generic definition of a number is that it is the cardinal property of a set.

Yes, I hear you already. "So why can't zero be thought of as the cardinal property of the empty set [?]" The answer is simple. If we understand the idea of a set as a collection of things, then zero can hardly be the property of a set. In order for it to be the property of a set we would need to comprehend the idea of a collection of things that is not the collection of a thing. That is an incompressible idea. Fool yourself as much as you like, but a collection of things that is not the collection of a thing is not a valid idea.

Mathematicians often try to get around this by claiming that the empty set is indeed a collection of a thing. It's the collection of nothing. Well, I have no intention of wasting everyone's time going down that road save to say that if nothing is considered to be a 'thing' then the empty set isn't empty after all now is it?

That might sound like so much hanky panky, but I tell you that it is genuine serious logic, and the problems that follow from such nonsense are insurmountable. Believe me, I've been down that road in agonizing detail, it's a dead end.

Trying to make something out of nothing is not the answer. It cannot work, and it's doomed to failure.

What's the other possibility? Well, we can agree that nothing is indeed genuinely nothing, and that the empty set is indeed empty. Now what?

Well, we originally comprehended the idea of a set as a collection of things. This was our comprehension. It was an easy concept to grasp. However, to now claim that we have a set which is not the collection of a thing we cannot help but ask? How do we comprehend this? Keep in mind that we are not permitted to simply think of it as the absence of a set. If we did that, it wouldn't be an empty set would it? It would be the absence of a set.

Nope, we MUST think of it in terms of our comprehension of a set as a collection of things. We cannot pretend that nothing is a thing, that will simply not work, it's a dead end remember?. So we are stuck with having to think of the empty set as something other than the collection of things that we had comprehended the idea of a set to be.

In math courses we would quickly be pushed beyond this logical problem and told to just shut up move on. Let's just go ahead and learn about operations on sets, etc, etc, etc. Everything will clear up in due time. And then we are shoved into category theory instead of quantitative theory.

As a pure mathematician that probability wouldn't bother me at all. I can live with category theory based on a bunch of incomprehensible axioms. The rules are the rules, and you just need to abide by them. Even if they are inconsistent as proven by Kurt Gödel.

However, as a physicist, I have to ask myself? Why am I fooling around with category theory when in reality I'm measuring, and describing, quantitative properties of the universe? Shouldn't I be using a quantitative based mathematics?

hurkyl wrote

Harry, Meg, and Joseph empty their pockets and place a bunch of coins on the table. Which of the coins on the table belong to George? The answer to this question is the empty set.

Obviously, since in category theory the empty set supposedly represents no quantity in this case. The empty set is the symbol used to convey this. It's just an axiomatic symbol. Accept it, don't try to comprehend it. That's category theory.

But if mathematics was a quantitative theory the answer would be simply zero. Yes, we would still use that same symbol, but instead of it referring to an empty set, it would be understood to mean the absence of a set, or more to the point, the absence of quantity.

By the way, mathematics actually can't even answer the question "Which of the coins belong to George?". The only question that mathematics can answer is "How many of these coins belong to George?". Mathematics is all about quantity (or at least it's supposed to be). The mere fact that we think that we can do other things with it simply emphasizes the fact that we view it as a category theory more than anything. It is becoming more and more a category theory, and less and less a quantitative theory.

Just for the record:

There are a lot of other consequences that follow from correcting set theory. It's not just a matter of how we think of the concept of zero. Many conclusions would change. Russle's paradox of the set of all sets, would be inapplicable, Gödel's incompleteness theorem would be inapplicable. Band aids would be falling off of set theory all over the place.

On Euclidian Geometry:

hurkyl wrote

But your system has an obvious problem; my notion of location may or may not be the same as your notion of location. For example, my notion of location would certainly permit the universe to have only one location (but empircal observation proves otherwise). The value of an axiomatic formulation is clear here; while our ideas about the fundamentals might differ, as long as we can agree that these fundamentals satisfy the axioms of, say, Euclidean geometry, we then have a common foundation upon which we can share ideas.

If we were working together diligently to develop such a system (instead of being at each other's throats about it) we could probably come up with a consensus on a comprehensible meaning for the idea of location.

I think that it is extremely important to keep in mind that if we were going to tackle the job of building a Euclidian geometry from scratch we would begin with the simplest of ideas (not start out with a 3-dimensional space that we haven't even invented yet!) We would begin with the simplest 1-dimensional case were locations would necessarily need to lie on a line before we even invented the concept. We would see the need for such new concepts as we progressed and define each of the new ideas comprehensibly as we progressed. It would be no small job. Let's give Euclid a little credit!

Surely you aren't going to try to tell me that you believe that axiomatic methods are the only possible way that humans can comprehend and convey their ideas with clarity?

I actually have no problem with axiomatic methods, by the way. I just have a problem with the one's that totally toss common sense and logic out the window! And one's the pretend to be about quantity when they are really about categories.

It's clear to me that you have been brainwashed (educated) to accept mathematics as a category theory. That's fine. But please don't try to tell me that it properly conveys and maintains the physical idea of quantity as we understand it. It absolutely does not. And the mathematical community has chosen this to be the case.

When they were looking for a formal definition for the Natural Numbers they insisted that any such definition must necessarily be removed from the idea of the thing that is being quantified. I mean, duh? What they heck were they thinking? Why remove the idea of number from the idea of quantity? To preserve some sort of lofty ideal of purity of thought? Why didn’t they focus on the 'thing' that they are quantifying? Why didn't they focus on the intuitive nature of the quantity of One?

It's really quite sad actually. :frown:
 
  • #13
NeutronStar
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lethe wrote:



well, i think hurkyl has this one pretty much in hand, but i guess i want to go down on record as saying that neutron star is way off base, and misses the point entirely of abstraction in mathematics, so take what he says with a grain of salt, that s my opinion, at least.

I'm glad that you brought up the topic of abstraction. This is a concept worthy of discussion.

One of the reasons that the mathematical community wanted to remove the idea of quantity from the things that are being quantified is because they believe that to do otherwise would somehow taint the pure abstraction of the idea of quantity.

Anyone who thinks about this deep enough will eventually break into a belly roll of laughter when they realize what the mathematical community actually achieved by doing this.

But first, let us ask ourselves what we mean by abstraction.

What does it mean for something to be abstract?

I could get out the dictionary and drown us in needless definitions, but I'd rather just look at some of the more important ones to save word space.

Abstract - a summary

I trust that everyone will agree that mathematicians did not intend to make mathematics abstract in the sense of merely being a summary of the idea of number.

Abstract - intangible

This is a good one. Yes, the mathematical community most certainly would want the idea of number to be applicable to intangible things. However, even intangible things are still things. So we can certainly attach the idea of number to the idea of a thing and still be well within an abstract formalism. All that is needed is for the thing that we use to be intangible itself. No problem there, and this is, in fact, precisely how a correct model of mathematical formalism would proceed. It would still be completely abstract in the sense of being applicable to intangible ideas of quantity.

Abstract - vague, unclear, difficult to understand

This is not a good one. Anyone in their right mind would avoid this type of abstraction in any formalism that they might attempt to construct. Yet the mathematical community seems to have embraced this one to the hilt. The whole idea of an empty set as a collection of things that is not a collection of a things is certainly vague, unclear, and difficult to understand. Yes, it most certainly wins the award for being abstract based on this definition of the word.

Abstract - applying to more than one case

Ah. This is my favorite. It is also the most important one as far as I'm concerned. Although, I do give some measure of importance to the intangible abstraction mentioned above as well. However, I see this definition of abstraction as the most important of all. Mathematics should indeed be applicable to more than one case. In fact, it should be applicable to any case where the idea of quantity is needed.

The changes that I propose to set theory would maintain mathematical abstraction in both, the sense of being applicable to intangible ideas of quantity, as well as applying to any case where an idea of quantity can be applied. As for the vague part, I'd just as soon skip that in my model of mathematical formalism.

So if anyone thinks that I'm missing any points about abstraction think again. I am quite capable of comprehending any ideas that are abstract in the sense of being intangible, or applicable to many cases.

However, I will agree that I do have trouble with abstract ideas that are unclear, vague, or difficult to understand. Especially when the person who is trying to convey the idea to me is having difficultly understanding the idea as well.

Yes, that latter type of abstraction completely eludes me. Can't imagine why?
 
  • #14
Hurkyl
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In mathematics, computer science, and probably other subjects as well, the process of abstraction is the process by which one distills the essence of a concept or idea from the surrounding hoardes of messy details that obfuscate the concept or idea.

This is not just a technical thing; people do it in every day life. For example, I don't need to know network protocols or hypertext markup language in order to surf the web. Even if I do know these details, I'm generally better off ignoring them while surfing.

Abstraction is often followed by generalization; once we've simplifed difficult concepts, one often becomes interested in other, similar concepts, or realizes the difficult concept can be applied to new problems.



Anyways, you seem fixated on the idea that the domain of mathematics should be limited entirely to numerical concepts, but it's not. There are different branches of mathematics, each considering different concepts; why should math do only numbers? And how does the fact some other branch of mathematics considers the ideas of shapes or sets affect the ability of other branches of mathematics to focus on quantitative concepts?


Some corrections.

Mathematicians do not use Cantor's set theory, they use ZF set theory.

Godel's theorem does not prove mathematics inconsistent, you should check the statement of the theorem again.

Godel's theorem needs only logic, the natural numbers, +, and *; any changes to set theory would have no bearing on Godel's proof.

You may have started down this road, but you got off well before the first signpost that says where the road leads. In particular, it may be true that all of mathematics can be built from set theory, but is patently false that such constructions are the essence of modern mathematics.

Category theory is (conceptually) a theory about objects with some structure and transformations that respect that structure; the term "category theory" the way you use it seems to have nothing to do with the mathematical theory "category theory".
 
  • #15
NeutronStar
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Hurkyl wrote:

Anyways, you seem fixated on the idea that the domain of mathematics should be limited entirely to numerical concepts, but it's not. There are different branches of mathematics, each considering different concepts; why should math do only numbers?

Yes, I am fixated on the idea that the domain of mathematics should be limited entirely to the concept of number (or more precisely the concept of quantity).

However, today it seems that any form of logic is now tucked under the umbrella of mathematics. When did this happen? And what about logic? Is logic itself still a valid discipline, or has it been swallowed up entirely up by the umbrella of mathematics? Does the term mathematics now apply to just anything at all that is logical?

Yes, as a physicist I am very interested in mathematics as a language and formalism to communicate and explore the nature of quantity. As a scientist using the scientific methods, measurement and quantity are extremely important to me. It's really the basis of my entire discipline. We live in a quantitative universe. It's principles of behavior are revealed by their quantitative relationships. So this is what I am interested in studying.


You may have started down this road, but you got off well before the first signpost that says where the road leads. In particular, it may be true that all of mathematics can be built from set theory, but is patently false that such constructions are the essence of modern mathematics.

Every and any meaningful number can be reduced to the cardinal property of a set. There are no exceptions.

I think that it's important to realize that any comprehensible idea of number can be reduced to the idea of a cardinal property of a set. This is in fact, our intuitive understanding of quantity, or better said, this is the nature of the properties of the universe that we call quantity. In fact, if a number can't be reduced to the idea of the cardinality of a set, then it is incorrect to call such an idea a number

Set theory most certainly is the very foundation of the idea of number. Whether formalized, or merely intuitive, this is the essence of number. So if at any time you are using a concept of number, then you are using set theory whether you realize it or not. The idea of number has no meaning outside of this comprehension.


Godel's theorem needs only logic, the natural numbers, +, and *; any changes to set theory would have no bearing on Godel's proof.

And this statement indicates to me that you are totally unaware of the meaning of number, and its connection to the idea of a set. To suggest that Gödel's proof needs only the natural numbers, and has no bearing on set theory is to suggest that the natural numbers have nothing to do with set theory. They are defined on it! This is the very concept of number. Change set theory, and you change the meaning of number. They are intimately connected and inseparable.

While it may be true that you can imagine using sets in ways other than quantitatively, it is not true that you can comprehend the idea of number without referencing the idea of a set.
 
  • #16
Hurkyl
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Scientists also have to describe the large-scale behavior of the universe. Describing ensembles is just as important, if not more important, than studying the fundamentals. Sometimes the large-scale behavior is still quantitative, such as currents or pressure... sometimes the large-scale behavior is strongly qualitative, such as the behavior of celestial objects.

Our methods of observation are quite numerical in nature, but generally numerical observation is a means to an end, not the end in of themselves.


Every and any meaningful number can be reduced to the cardinal property of a set. There are no exceptions...

I'm curious what you mean by this; do a nontrivial example (i.e. not a natural number)... maybe pi or e.


To suggest that Gödel's proof needs only the natural numbers, and has no bearing on set theory is to suggest that the natural numbers have nothing to do with set theory. They are defined on it!

No, the natural numbers are (typically) defined through peano's axioms. Godel's theorem uses no knowledge of the natural numbers beyond the consequences of their arithmetic.


it is not true that you can comprehend the idea of number without referencing the idea of a set.

I don't have a problem with it.
 
  • #17
NeutronStar
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Hurkyl wrote:
I'm curious what you mean by this; do a nontrivial example (i.e. not a natural number)... maybe pi or e.

I knew that you were going to come back with the concept of the irrational numbers.

Actually I'm writing a book about this. Thus far we have been discussing concepts that I talk about in chapter One of my book. I cover the phenomena of irrational numbers in chapter 7. However, there is a lot of critical information in the chapters in between. Not the least of which is a chapter on properties of self-referenced quantities.

As you must know both pi and e are numbers that arise from self-referenced quantities. Pi is obviously the self-referenced situation of the ratio of the diameter and circumference of the same geometric object. Change one and the other must also change as these two quantities are directly connected to the same object (they are self referenced).

One of these two quantities must be irrational (either the diameter, or the circumference). We can chose to push the irrational property around so to speak. A circle with a diameter of 1 for example has an irrational circumference. A circle with a circumference of 1 must have an irrational diameter. It's like a squeezed balloon. We can push the irrationality around, but we can't make it go away.

The square root of 2 is another irrational number. But it's a magic one! It has a very special property. It shows us clearly that two irrational conditions can be used to make irrationality go away. Take the square root of 2 (an irrational number) and multiply it by another irrational number (say, the square root of 2), and what do we get? The perfectly rational number 2.

I think that's a pretty slick trick actually.

It shows that irrationality can be quite rational in some situations.

The square root of 2 is also a situation of self-reference (as are all meaningful irrational quantities). The square root of 2 is the number that when added together precisely the same amount of times that it represents it will equal 2. That's self-reference. We are adding a quantity to itself precisely the same amount of times that it represents.

Of course the irrationality of the square root of 2 can also be pushed around like a squeezed balloon in much the same way as pi. Instead we simply use a right triangle. We can either make the sides irrational, or we can make the hypotenuse irrational. But we can't make the irrationality go away from that self-referenced situation.

Euler's number e is also an example of self-reference. I'm not going to type in the details. You are obviously mathematically literate enough to recognize the self-referencing of that situation on your own. If you have trouble let me know and I'll explain it in a later post.

In short, irrational numbers are a special case. This doesn’t mean that we can't comprehend them in the sense of cardinality. We most certainly can imagine their finite approximations as a cardinal property of a set. And ironically there is no limit to how far we can carry those approximations out! Just as long as we keep them finite.

That almost doesn't make sense, but it actually makes perfect sense. There are many things in mathematics that we can carry out without bound, but we must still keep them finite.

In fact in chapter 6 of my book I explain why their can only be a finite number of points in a finite line. You can probably imagine the abstract consequences of that. I prove it using sound logic and show that it's really no different than the infinite set of finite natural numbers that we have been accepting for years.

There are a lot of other interesting things going on in the other chapters as well, and to really get an appreciation of the later concepts it would pay the reader to follow it one concept at a time.

In any case. I didn't come here to rewrite my book on an Internet forum board.

I actually just wanted to point out some truths about mathematics to On Radioactive Waves. If he is interested in the foundations of mathematics he should be aware that set theory is indeed the very foundation of the idea of number. And despite what many abstract mathematicians might believe, set theory is indeed the foundation of all of mathematics. I can site several mathematics textbooks that state as much (not the least of which is the "Foundations of Higher Mathematics" which I have already given the ISBN number for).

To answer your question in short, no, irrational numbers cannot be comprehended in their entirety, as a cardinal property of a set. However, you have to understand that this is not a problem at all. They can still be approximated to any precision desired, they can still be taken to the calculus limit as always. And their approximations can indeed be thought of as the cardinal property of a set. The only final result of such a formalism would be to simply recognize that they technically don’t qualify as numbers by definition. At least not in their entirety. They would still qualify as numbers when approximated to any desired degree of accuracy.

I don’t see where this would be the slightest problem. We can only use them in their approximate forms anyhow with current mathematics. So what's the big deal?

The important thing to realize that they don’t satisfy our intuitive idea of quantity. And that is actually quite valuable information from a physicists point of view. It tells the physicists that the things that we are trying to quantify when using irrational numbers cannot be quantified using our intuitive understanding of the concept.

Wow! To me that is some really powerful stuff! And if you look at where we these numbers arise you will discover that this is indeed true. There is no quantitative unit of space in the universe. Space is a relative concept. There is no absolute space. Of course, we already learned that from Albert Einstein, but mathematics was telling us this all along when we tried to quantify space and got irrational numbers. Instead of paying attention to what it was telling us, we just force the concept of an irrational number onto the formalism of mathematics. When will we ever learn?

I really do need to quit posting my ideas to this board. They don't seem to be appreciated anyway, and I could really use this time to finish my book. Then I can just publish it and instead of arguing each point I'll just post the link where people can buy the book.

I'll hang around on this site though. I might need some help in my quantum mechanics homework. I may have a real good understanding of set theory and some of mathematics, but I could still use some help with differential equations, and Bell's Theorem which I find to be one of the most interesting problem in physics.

Thank you all for putting up with me, and to, On Radioactive Waves, I hope you find the information that you are seeking.

And Hurkly, thanks for putting up with me. I've read some of your posts and you do seem to have a very good understanding of physics. Probably the only thing holding you back from discovering a unified theory of everything is the flaws of the pure abstract mathematical formalism that you are using. I think that you will find it next to impossible to discover the true nature of a quantitative universe whilst using a qualitative mathematical formalism.

But hey, I could be wrong. I was wrong once before. I think?
 
  • #18
HallsofIvy
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"The square root of 2 is the number that when added together precisely the same amount of times that it represents it will equal 2."

Could you tell HOW you are going to add square root of 2 (or ANY number to itself) "square root of 2" times? I don't see how you can use anything but a counting number to count.

You seem to be using the term "self reference" in a way different from any definition I have seen before. Certainly the fact that
square root of 2 is defined by "square root of 2 times square root of 2= 2" (and square root of 2 occurs twice in that definition) does not mean it is "self-referential".

Finally, irrational numbers are a "special case"? How special can they be- almost all numbers are irrational.
 
  • #19
On Radioactive Waves
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Thanks for your detailed postings Neutron Star. Yes, this thread has given me much (although not all) of the information I was looking for. I know it's my job to find what I'm looking for, and in fact this thread had a better response than I expected. I didn't get to finish reading your last post yesterday, but I finished now.

Here is something I don't understand about your viewpoint.


In fact in chapter 6 of my book I explain why their can only be a finite number of points in a finite line. You can probably imagine the abstract consequences of that. I prove it using sound logic and show that it's really no different than the infinite set of finite natural numbers that we have been accepting for years.

How do you conclude this? My understanding of a point is it has the property of location, but has no dimension. A line has dimension of length. The two arn't dimensionally equivilant. could you elaborate?
 
  • #20
Hurkyl
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It seems you are basing your justification on a physical interpretation of mathematics; such a justification only applies to the choice of mathematical theories used to describe physical reality, not the internal logic of mathematics itself.


Probably the only thing holding you back from discovering a unified theory of everything is the flaws of the pure abstract mathematical formalism that you are using.

Well, I think the main thing holding me back is I went into the fields of mathematics/computer science and am only a hobby physicist. But I do make the effort to have more than a layperson's understanding of things, and am even studying to learn things the "right" way. (for instance, I'm participating in the thread on Lie Groups, and high level continuous mathematics is unlikely to have much application in my field of work)


Anyways, the same objection I had with pi and 3 works with 1/2 and -3. The purely set theoretic constructions of anything but the natural numbers requires something beyond the notion of a cardinal number; they actually require using sets as sets instead of ways to generate cardinal numbers; the identities of the objects contained in those sets are important.


And we do use the irrational numbers in their exact forms, even in some computational domains (see Mathematica).
 
  • #21
NeutronStar
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On Radioactive Waves wrote:
How do you conclude this? My understanding of a point is it has the property of location, but has no dimension. A line has dimension of length. The two arn't dimensionally equivilant. could you elaborate?

Your understanding of a point as a location is perfectly correct. This is all that it is, and for this very reason it is indeed dimensionless.

But that's only a single point. Any two points (which are not the same point) must necessarily be associated with a distance (the distance between them).

When we are talking about the points in a finite line this distance between the points cannot be ignored (neither can it be made to go away). After all, to claim that it goes away is to claim that adjacent points are "touching". That certainly can't work! Points are nothing more than dimensionless locations right? So if they are said to "touch" then they would necessarily be the same point. This certainly isn't going to do the concept of a finite line any justice.

So now we have a sound argument as to why there must be a finite number of distances in a finite line. And of course, if we have a finite number of distances then we must also have a finite number of points defining the ends of these distances.

That logical argument right there is actually good enough for me.

However, I go much further than that in my book. I prove, using set theory, why it is impossible to force an infinite number of points into a finite line. I will reserve that proof for my book.

After all, On Radioactive Waves, I didn’t come into this thread to prove this piece of information, I merely mentioned that I do so in my book.

For the sake of protecting that information let me just say that I believe I have also proven it with set theory. Perhaps I could be wrong. When I publish my proof you can read it and let me know what you think. :wink:

I will give you this much information.

The two key elements of the proof are this:

1. There necessarily must be a finite distance between any two points in a finite line. (An absolute truth)
2. The quantitative property of a set does not automatically transfer onto it's individual elements. (An absolute truth)

Using these two truths as a hypothesis I show that a finite line can only contain a finite number of points. It is a logically correct proof. It's also a very simple one to follow. Quite eloquent if I must say so myself. :smile:

As a final note, the set of natural numbers has this identical same property, and this comparison brings it all home crystal clear. Not to imply that there are finite many natural numbers, on the contrary there are indeed infinitely many. The comparison follows more along the lines of 'key element number 2' above.
 
  • #22
NeutronStar
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HallsofIvy wrote:
Could you tell HOW you are going to add square root of 2 (or ANY number to itself) "square root of 2" times? I don't see how you can use anything but a counting number to count.

Excuse me? Are you Hurkyl's twin brother or something? Or is this just Hurkyl's second screen name?

Anyhow, whoever you are I'll answer your question.

What's 2*4? That's 8 isn't it? How about 3*4? That's 12 I think? And 4*4? 16?

How did we get those answers?

Let's see 4+4=8. Hmmm?

How about 4+4+4=12. Hmmm?

And 4+4+4+4=16. Hmmm?

Seems like some sort of pattern going on here?

Could it be that multiplication is just a shorthand notation for repeated addition?

I wonder if we can take this to a bit more abstraction and imagine multiplying numbers that aren't whole?

1.5*1.5? That gives 2.25 if I'm correct.

What happens if we use the concept of repeated additions here?

Well add it once that's 1.5 (we started with the absences of a set)

(Or for those who like the empty set we can say that we added it to the empty set the first time)

Well, now we must add half again as much to complete the repeated addition right.

Half of it would be 0.75 so we add this to the 1.5 and we get 2.25.

Wow! Would you look at that! Abstraction at it's finest!

Multiplication and repeated addition are the same thing.

By golly! I knew that, I swear I did!

So what was the question again? Oh yeah, we want to see the self-referencing associated with the square root of two?

Well, now that we understand that multiplication and repeated addition are one and the same thing, we can clearly see that the [squ]2 is the number that when added together the same number of times that it represents adds up to 2.

This is a form of self-reference. The number that is being added together is the very same number that is controlling the repeated addition. It's like pulling up it's own bootstraps.

If you have trouble completing this repeated addition see Hurkyl for the calculus of series summations, (unless you are Hurkyl), then you obviously already know how to do that.
 
  • #23
NeutronStar
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Originally posted by Hurkyl
Well, I think the main thing holding me back is I went into the fields of mathematics/computer science and am only a hobby physicist.

Well, this explains a lot. Being that this is a physics forum I naively assumed that you are more physicist than mathematician. Realizing that it is the other way around make me respect you all the more actually. (Not for your abstract mathematics. ), but for your impressive knowledge of physics. I've met actual physicists that don't seem to have a grasp of some of the concepts that you seem to understand rather well.


Anyways, the same objection I had with pi and 3 works with 1/2 and -3.

Actually, in my book I propose a whole new way of looking at set theory. It really isn't all that different than current mathematics actually. I mean, there are some major difference, but most of them are conceptual rather than practical.

For example. My number line begins with 1 and goes off toward infinity. It doesn't begin with zero. Zero is not a number in my system of mathematics. It's still a valid concept that means the absence of a quantity, or absence of a set, and in a practical sense the symbol is used in almost the precise exact way. It simply represents a different concept than the empty set.

Now, since my number line begins with 1 this will probably have you asking what happens to the rationals. Well, I still use them in almost the very same way that current mathematics uses them. The only real difference is that I conceptually keep them in the form of the ratios, divisions, or the repeated subtractions that they represent. I treat them almost like numbers, but I always understand in my mind which concept they represent.

I see division in two distinctly different ways. One I call "dividing up the pie" where it is absolutely necessary to create new units to go with the quantities. The other one I simply see as repeated subtraction where new units are not necessary.

I put the word units in italics above, because in my mathematical formalism it is not possible to talk about a quantity that is not associated with a thing. The concept of a pure number without something to quantify is senseless. So every number in my system must be associated with a thing that is being quantified. If you get the impression that somehow denies abstraction think again. It does not. I have yet to find a concept in our current mathematics that I cannot associate with some type of abstract individual thing. And the most important property here is that of "individuality". If an object cannot be shown to have a property of individuality, then you cannot claim to quantify it using the concept of number. This is really the major correction to mathematical formalism. As it is now, it is quite acceptable to quantify things that do not have a property of individuality. Tsk tsk. That will definitely lead to a paradox or logical contradiction.

Finally on this topic. I also do not permit absolute negative numbers. The property of negativity is reveal to be a relative property between sets and not an absolute property of a set. Not so with current mathematical formalism. They view negative numbers as absolute sets. Again, tsk tsk! That leads to problems because it's not an absolute property, it really is a relative property of quantity. Current mathematics deals with this only superficially by recognizing the absolute value function. In my mathematics, I still use the absolute value function, and I still us what appear to be negative numbers in almost an identical way. The only difference is that in my formalism it is recognized that the negativity of a quantity is relative between sets, and not an absolute property of the concept of number. No negative numbers can stand alone in my mathematics. They would be totally meaningless concepts as such.


And we do use the irrational numbers in their exact forms, even in some computational domains (see Mathematica).

Well, yes and no. I mean, many computer systems can carry the symbols of irrational numbers around. My TI-92 Plus will do that. Be it can't use the value of pi in it's entirety in a numerical calculation. Surely no computer can do that. Unless it is just using a calculus trick in the background to return some rounded value of a limit? I mean, how could any finite computer compute pi to an infinite number of places?
 
  • #24
Hurkyl
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Well, this explains a lot. Being that this is a physics forum I naively assumed that you are more physicist than mathematician. Realizing that it is the other way around make me respect you all the more actually. (Not for your abstract mathematics. ), but for your impressive knowledge of physics. I've met actual physicists that don't seem to have a grasp of some of the concepts that you seem to understand rather well.

Bleh, you're giving me a big head. :wink:


Zero is not a number in my system of mathematics...It simply represents a different concept than the empty set.

Such as the additive identity?


I treat them almost like numbers, but I always understand in my mind which concept they represent.

Rational numbers are different from natural numbers... I'm not sure what you mean by this point except you don't like to use the term "number".

Incidentally, a typical construction of a fraction field from a domain (such as the rational numbers from the integers) is simply to take the set of ratios.


I put the word units in italics above, because in my mathematical formalism it is not possible to talk about a quantity that is not associated with a thing. The concept of a pure number without something to quantify is senseless.

I presume that arithmetic is the same no matter what unit is attached? I.E. if x apples + y apples = z apples, then x oranges + y oranges = z oranges?

With this sort of formalism, it is still possible to talk about dimensionless numbers in several ways; I presume it isn't fair play to just drop the dimensions from numbers, but one can

(a) Use modulo arithmetic... but instead of the modulo being something like 12, the moduli are 1 X - 1 Y for any pair of dimensions X and Y.

(b) Use the isomorphisms between the arithmetic of different units, so pick some unit as "special", such as apples, and then convert all numbers into apples, perform the arithmetic, then convert back

(c) Consider ratios; the ratio between, say, 2 apples and 1 apple would be dimensionless, and yields an arithmetic equivalent to traditional mathematics.


From a foundation such as yours, dimensionless numbers still serve a purpose; it allows you to study the form of arithmetic... and then apply the results of your studies to dimensional computation.


They view negative numbers as absolute sets.

What do you mean by "absolute sets"? Negative numbers are those numbers of an ordered (additive) group that are nonzero and nonpositive. (alternatively they are the numbers of an ordered (additive) group less than zero; the additive identity)

Of course negative integers may be thought of as particular differences of natural numbers (and is precisely how the set theoretic construction of the negative numbers works).


I mean, how could any finite computer compute pi to an infinite number of places?

We don't always need to use decimal expansions to do computations.



fter all, to claim that it goes away is to claim that adjacent points are "touching". That certainly can't work! Points are nothing more than dimensionless locations right? So if they are said to "touch" then they would necessarily be the same point. This certainly isn't going to do the concept of a finite line any justice.

Concepts of nearness require one to consider sets of points to be very interesting. Two points can't be near each other unless they're the same point (at least in non-pathological topological spaces)... but a point can certainly be near a set of points without belonging to the set.

(I'm not using "near" in a vague way; it has a precise topological meaning)

It's easy enough to have an infinite number of points on a finite line; one doesn't even need to go with the full force of the "continuum"; there are an infinite amount of rational numbers packed into the range [0, 1].


PS: claiming "absolute truth" is not a good way to win followers; any sort of logical proof is necessarily relative to your hypotheses.
 
  • #25
NeutronStar
419
1
PS: claiming "absolute truth" is not a good way to win followers; any sort of logical proof is necessarily relative to your hypotheses.

True. I should have said, "self-evident truth". Then if there is a reason why someone feels that it isn't evident we can compare our understandings of the concepts to see why it seems evident to one person and not to another.

Sometimes there can be loopholes or other things that the first person missed. I'm obviously pretty confident that I've covered all the loopholes. But just like Von Neumann's proof that no hidden variables can exist in QM, you have to first accept his hypotheses. So I guess I was really giving my so-called "absolute truths" as the foundation of my hypotheses. Reject either of my hypotheses and of course my proof falls flat on it's face.

I'm a lot better at thinking through these proofs than I am about popping them off on an internet forum board.

I do take better care of the wording of such things in the actual book that I am writing. I also have proof readers who point some of these things out (although they would not have caught the difference between an "absolute truth", a "self-evident truth", or a mere postulate for a hypotheses. Although in this case I think that "self-evident truths" would be a better fit than to claim that they are "postulates". To say that they are postulates sounds like I just made them up on my own. I think they have a little more creditability than that.

(c) Consider ratios; the ratio between, say, 2 apples and 1 apple would be dimensionless, and yields an arithmetic equivalent to traditional mathematics.

This is the kind of thing that never ceases to upset me in physics classes. From my point of view there is no such thing as a dimensionless quantity. The example that you just gave here is 2 apples per 1 apple. So we see that the common unit of apples can be canceled and we just settle for calling it a ratio of 2 without dimensions (or a ratio of 1/2 if we are looking at it from the other point of view).

But in every practical case of a meaningful ratio there is always a "reason" why the ratio make sense. 2 of John's apples to every one of Mary's apples. Or maybe 2 green apples for every red apple. In other words, there is always more information available in the units that we are disregarding or tossing out when we cancel units.

I remember one time in a physics class I was given a "pure" number as a constant in an equation. I was trying to make sense of the whole situation. I wish I could remember the precise problem but I can't. In any case, I was totally baffled to figure it out. Finally in frustration I went to my professor and asked what the dimensions were of the so-called dimensionless constant. At first he kept insisting that it doesn't have dimensions, but with persistence I was finally able to discover what the units were that canceled. That information was enormous help to me because it exposed the other force, energy, or whatever it was that had supposedly canceled out. And that knowledge helped my understand what was going on. The ratio of force per force, or energy per energy made sense to me because I could comprehend it in terms what was going on in a physical picture in my mind. From that day on I swore that I would never accept another so-called dimensionless number.

Even pi can be said to be a dimensionless ratio, but it's not, it's distance per distance, but by golly when you know who those distances belong to it makes a world of difference doesn't it? It's the distance of circumference per diameter. Now those units don’t cancel!


Did I state that correctly? I have lysdexia and I often state things backwards.

There may be other ways to describe pi too, but in all cases it will be a ratio of two different concepts. Those two concepts will always be connected in a self-referenced situation by the way. :wink:

If they aren't you either missed the connection, or I want to know about it! :smile:

I would actually love for someone to give me a meaningful irrational quantity that isn't a result of self-referencing. I would LOVE to be proven wrong!

Infinite strings of meaningless numerals are totally uninteresting. I don't see that as a meaningful quantity anymore than I would see an arbitrary string of alphabet characters as a meaningful word.
 
  • #26
Hurkyl
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To say that they are postulates sounds like I just made them up on my own. I think they have a little more creditability than that.

I've seen quite enough times where a person thought something was this obvious, yet were totally mistaken (including myself) that I loathe to accept anything as "self-evident". :wink: (though I do occasionally use the term for the sake of expediency)


I remember one time in a physics class I was given a "pure" number as a constant in an equation.

I know about this; it comes in two flavors... one flavor where you might get an equation like "the position of the object is given by 3t2 + 4t - 6". I utterly detest this because it's often done in contexts where the student is supposed to be carefully keeping track of units in every formula... making this equation meaningless in such a context!

The other flavor seems to be of what you speak, though... I have invented more detailed units on my own to solve problems like this in the past!


The problem with this programme is that it only seems to be applicable on a problem by problem basis. Sometimes "10 of John's apples" and "5 of Mary's apples" would be useful for the problem... but it becomes a syntactic nightmare to write with the same precision a formula saying how many apples I now have after John and Mary give me all of their apples.

I had been preparing a ratio problem to demonstrate this difficulty as well: If I have 5 oranges and 10 apples, and then someone comes along and gives me n copies of each object I have, and I inspect my oranges and I have 15, then how do I compute the number of apples I have, with complete rigor?


I think such an ambitious dimension tracking scheme requires everything to be placed into an extensive type hierarchy; there is a difference between John's apples and Mary's apples, but at the core of things (pun intended!), they're all apples. When doing problems, the notation would have to slide up and down the hierarchy to choose the appropriate syntax for the problem, and at this point it really seems that the point is merely to be a notation to assist solving problems instead of a useful rigorous requirement on all equations.

And it seems practical to have some very general things like "countable object" or "measurable concept" way up near the top of the type hierarchy, and it also seems practical to at least allow the existence of a multiplicative identity (i.e. "dimensionless"), even if it is only useful on rare occasion.


BTW, your explanation of pi as self-referential seems to be a big stretch; it is the ratio of two totally different things (a circumference and a diameter) that happen to be related by a higher concept (geometry of circles).
 
  • #27
NeutronStar
419
1
Hurkyl,

I want to thank you for the really great discussion. I hope you'll be around this fall to help me with my homework problems. :wink:

Right now I simply must refrain myself from spending so much time typing my ideas into a public Internet forum. My real interest is in QM and not mathematics anyhow. I kind of got drawn into the whole mathematics thing from studying GR actually. Just for kick I'll outline the road that took me to the discovery of the problems associated with the empty set.

On Radioactive Waves: You may also find this interesting,…

I was reading about General Relativity (not a serious study program) when I ran across an idea about inertia given by Ernst Mach. Mach's principle, as it was called, is that all of the matter in the universe collectively acts to produce the effects of inertia. Einstein liked the idea, so they tried to work this into GR.

Along came a fellow by the name of Kurt Gödel, a pure mathematician. He proved, using pure mathematics, that Mach's principal was incompatible with GR (something to do with the fact that distant galaxies spin). I didn't quite understand Gödel's proof, and I was highly suspicious of a pure mathematician proving something in physics, so I began to study Kurt Gödel himself. This lead me to his Inconsistence Proof, and that in turn lead me to the definition of the natural numbers, and the operation of addition as defined on set theory.

It took me quite a long time to figure out what the problem was, but I finally realized that it all came down to the property of individuality. Or perhaps I should say, the lack of the property of individuality of mathematical objects in mathematical formalism. And this ultimately lead me to the empty set and Georg Cantor. This was the end of the road. Or perhaps I should say the beginning of my road to build a new set theory completely with a proper quantitative property of individuality at its foundation. The rest will be history when I finally publish my book. :wink:

By the way Hurkyl, I do have a fundamental abstract "countable object" in my formalism. I'm not going to even being to attempt to define it here because there are so many old issues with the empty set that need to be addressed first in order to properly understand the new concept. I actually don’t formally define it until chapter 3 in my book. And I cover a lot of ground in both chapter 1 and 2 prior to that.

Anyhow, I thought I'd just give that quick overview before I leave in case anyone finds it interesting.

Hurkyl worte:

I had been preparing a ratio problem to demonstrate this difficulty as well: If I have 5 oranges and 10 apples, and then someone comes along and gives me n copies of each object I have, and I inspect my oranges and I have 15, then how do I compute the number of apples I have, with complete rigor?

I'm afraid that I don't see the problem here? I would simply see 'n' as having the units of copies and with my understanding of that unit I would have no problem formally completing all operations required. When I was finished with the operations I would end up with only apples and oranges, (no copies left). :smile:

See, there's just too much to explain. In chapter 5 of my book I explain mathematics as a language. In that chapter it would become clear how copies turn into apples and oranges. It's not really necessary to drag all these units around with us when actually doing calculations. It is only necessary that they be understood and well-defined within the formalism itself as a whole.

Here's the Table of Contents of my book,…. Then I must go. Sorry. :frown:

1. What is mathematics? - explains the ideas of number and set theory from an intuitive point of view.

2. Starting With Nothing - explains Cantor's empty set theory in current mathematical formalism

3. Starting With Some Thing - explains my new set theory giving credit to Guissepe Peano.

4. Applied Mathematics - explains all elementary operations of numbers in terms of the new set theory

5. Communicating with mathematics - explains how numbers are nouns, operations are verbs, and signs like negative, positive, and others are adjectives. This chapter clears up much about what actually goes on during operations (like the changing of copies into apples and oranges)

6. What's the Point? - This entire chapter is dedicated to showing that a finite line can only contain a finite number of points. We come out of this chapter with a whole new appreciation for set theory and infinity.

7. Other Interesting Stuff - Imaginary numbers, Prime numbers, Irrational Numbers, Infinity and fractals
The imaginary numbers are shown to be just normal cardinals - the concept of mathematics as a language helps here with the verbs, and adjectives. The imaginary part of an imaginary number is just an adjective, like plus or minus, only slightly different.

8. The Real World - This chapter focuses on some interesting applications in physics.

9. Summary - this small chapter ends the book with an overview of all of the major differences between a mathematics based on nothing (the empty set), and a mathematics based on some thing (an object with a property of individuality). While much is almost identical, there are some significant differences.

I must quit posting now, because I can't afford to put anymore time into this. However, I will remain on this forum, and I may post a short comment or two on other threads, maybe actually about physics for a change. :wink:
 
  • #28
OK, I'm pretty new to set theory, I just want to check I have this right.

Ø = An empty set
{Ø} = A set containing an empty set
{Ø,{Ø}} = A Set containing an empty set and a Set containing an empty set.

|Ø| = 0
|{Ø}| = 1
|{Ø,{Ø}}| = 2
correct ?

Are there any other differances between Ø, {Ø} and {Ø,{Ø}} other than the differences stated ?
 
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  • #29
NeutronStar
419
1
JAGGED wrote:

Are there any other differances between Ø, {Ø} and {Ø,{Ø}} other than the differences stated ?

From a purely editorial point of view I would read Ø, {Ø} as the list of numbers 0, 1. And I would view {Ø,{Ø}} as a set with a cardinal property of 2. (Or simply as the set containing 0 and 1)

The problem associated with the distinction between an element and a set is really fundamental to the very idea of defining the number 1 based on the idea of an empty set. The importance of this distinction isn't going to be very apparent for numbers greater than 1. However, the logical need for a phantom property of a set has already been generated at this fundamental level. And it does come into play when talking about concepts of infinity. In fact, it is this very property that permits Georg Cantor to create sets that are larger than infinity. Take away this phantom property and all of a sudden there is only one infinity! The idea of infinities greater than infinity are no longer possible.

By the way, since you are just starting out let me tell you where it all ends up!

Cantor ends up proving that the set of real numbers is more infinite than the set of natural numbers!

Actually he goes on to prove a whole bunch of stuff about infinities larger than infinity.

I don't accept his proofs, and in my book I point out to be what I believe to be logical flaws in his proofs. In my formalism there is only one concept of infinity. It is the idea of endlessness. We either have this condition or we don't. In my formalism it is impossible to prove that something is more endless than something else. A set either has the property of being endless or it doesn't.

Cantor's proofs are based on a principle of One-to-One correspondence. The fact that he has ill-defined the very concept of One as a subjective qualitative idea rather than as an objective quantitative idea is what permits him to get away with his silly proofs. I do not accept them, and I denounce their logic as being contradictive.

Now look at me! I promised to stop posting and here I am! This is addictive! I need to seek out professional help!

It seriously wasn't my intent to get this deep into it. I was merely trying to advise On Radioactive Waves that current mathematical formalism is not as sound as mathematicians will lead you to believe. After all, they aren't about to openly admit to newcomers that they have problems with their discipline! But behind the scenes they are well aware that these problems exist.

And on a final note, I do believe that all of mathematics (at least any mathematics that has to do with the concept of number) is absolutely based on set theory. This *is* the foundation for the idea of number and therefore it is the foundation for any part of mathematical formalism that refers to the idea of number. As far as I'm concerned the parts that don't rely on the idea of number have no business being classified as mathematics. They are simply other forms of logic. Why call them mathematics? Just because they are taught in the mathematics departments of educational institutions?
 
  • #30
Hurkyl
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Are there any other differances between Ø, {Ø} and {Ø,{Ø}} other than the differences stated ?

Nope! (except, of course, all the differences that can be expressed in terms of the stated differences)


From a purely editorial point of view I would read Ø, {Ø} as the list of numbers 0, 1. And I would view {Ø,{Ø}} as a set with a cardinal property of 2. (Or simply as the set containing 0 and 1)

I would like to point out that the cardinality of a set does not contain the entire picture. For example the set

{ {{Φ}}, Φ}

is different from

{ {Φ}, Φ}

although they both have cardinality 2. Or for a more obvious example:

{Apple, Banana} is different from {Orange, Banana} even though they both have cardinality 2.


In fact, it is this very property that permits Georg Cantor to create sets that are larger than infinity.

[?]

I'm hard pressed to see how powersets, functions, and the diagonalization argument have anything to do with what you said.


And I'm curious what is flawed in the meaning of 1-1 correspondence? I don't see what your objection to the "meaning" of the number one have anything to do with 1-1 correspondence except they happen to have the same word in their names.


In my formalism there is only one concept of infinity. It is the idea of endlessness. We either have this condition or we don't. In my formalism it is impossible to prove that something is more endless than something else. A set either has the property of being endless or it doesn't.

So? That, in of itself, doesn't mean two endless things have the same size.

And just because you don't like the choice of words to describe the idea doesn't mean the idea does not have merit.


It seriously wasn't my intent to get this deep into it. I was merely trying to advise On Radioactive Waves that current mathematical formalism is not as sound as mathematicians will lead you to believe. After all, they aren't about to openly admit to newcomers that they have problems with their discipline! But behind the scenes they are well aware that these problems exist.

And I'm trying to advise him that these problems are in the eye of the beholder; people who don't like a particular way of thought often see problems in it that don't exist.
 
  • #31
NeutronStar
419
1
Hurkyl wrote:

I would like to point out that the cardinality of a set does not contain the entire picture. For example the set

{ {{Ø}}, Ø}

is different from

{ {Ø}, Ø}

although they both have cardinality 2. Or for a more obvious example:.

I think JAGGED was using the formal definition of 2 as defined by Cantor.

Using Cantor's notation for the Natural Numbers the set { {{Ø}}, Ø} would actually be undefined. Although it could abstractly be thought of as the set of some object and zero, which would indeed have a cardinality of 2. There is no Natural Number defined as {{Ø}}.

And while it's true that { {Ø}, Ø} also has a Cardinality of 2, It is usually written the other way around just as a convention. Cantor's set theory does not require any particular order to the elements in his definition of the Natural Numbers. Cantor just wrote it out in the conventional style of increasing order of quantity for the sake of clarity. {0, 1, 2, 3, 4, …..} only using the formal set theory definition for each number in the list.

Cantor's set theory is actually dependent on a qualitative idea to force the idea of quantity. In other words, based on Cantor's logic a set with a cardinal property of 2 must necessarily contain items that are qualitatively different from each other. From the purist philosophical notion it is impossible to have say, 2 basket balls, according to Cantor, but cause these objects are identical they can only be viewed as one object.

Of course, we simply ignore this formalistic detail when we apply mathematics to real world situations. It seems very ironic to me that we can so quickly ignore the formalism of mathematics when actually applying it, and yet hold so steadfast to the idea that it is somehow a sound abstract logical system. It is not.


I'm hard pressed to see how powersets, functions, and the diagonalization argument have anything to do with what you said.

And I'm curious what is flawed in the meaning of 1-1 correspondence? I don't see what your objection to the "meaning" of the number one have anything to do with 1-1 correspondence except they happen to have the same word in their names.

The idea of a 1-1 correspondence is just that, a correspondence between 1 particular thing, and 1 other particular thing. These things are always elements of sets, whether formally recognized as such or not.

So they don't merely have the same word in their names. One means One for crying out loud! What are you suggesting here? That we can use the word One to mean whatever we choose? (we actually do this in applied mathematics as I pointed out earlier, but in that case we are just ignoring the incorrect formalism and acting on our correct intuition of what we know number to mean)

The idea of One is defined in formal mathematics as the feeble quantitative idea of set containing the empty set.

That's it period! There is no other definition for One! If there was it would be an ambiguous concept. When talking about the idea we would need to constantly be asking each other, "Which meaning of One are you talking about?" We don't need to do that because the concept of One is defined by Georg Cantor's set containing the empty set. That's it. That is the definition of 1.

There are no other formally accepted definitions of One that I am aware of. And if there are, then mathematics would even be more ambiguous than I currently believe!

So? That, in of itself, doesn't mean two endless things have the same size.

And just because you don't like the choice of words to describe the idea doesn't mean the idea does not have merit.

What?

It sounds to me like you are indeed thinking of infinity as representing some type of finite idea of quantity. Or course, I realize that you are not alone, most pure mathematician do view it this way, and Cantor's larger and smaller infinities only serves to encourage this ridiculous notion.

Endless things don't have a size, they are simply endless.

If two endless things are both endless then that's what they are. It's absurd to say that one is more endless than the other.

Now, I will grant you that things that are quantitatively endless may indeed have distinctly different qualitative properties. No problem there. But if that's the case we should clearly recognize that these separate and distinct qualitative properties are not an idea of quantity. For this reason mathematical formalism should not even attempt to address them.

We can still address them using other formalisms of logic, but pushing this job onto mathematics only serves to further muddle the idea of quantity. And this is one of my biggest concerns.

In my mathematical formalism infinity represents a quantitative concept of endlessness. That's it. It represents a set which contains an amount of elements that cannot be expressed as a finite number. That's it. That's what it means. Period amen. No ambiguity. No infinities that are larger or smaller than any other infinities are permitted. A set either contains a finite number of elements or it doesn't. There's simply no other choices possible. In my formalism it would be absurd to claim that any infinite set is any more infinite than another infinite set. The definition is clear and precise. A set is either infinite or it's not. There simple isn't any other choice available. It simple, clear, and unambiguous. Not to mention that it is a totally comprehendible concept.

By the way, Cantor's diagonalization argument doesn't hold water. It's a complete sieve, and I reveal it's fallacy in my book.
 
  • #32
Joy Division
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Just out of curiousity, are you working from Cantor's original work?

There has been progress in set theory since then I wonder if you've looked into that?
What about Bertrand Russels category theory? That was one competing notion for the foundations of mathematics, it didn't work though and Cantor's set theory did *shrug*.

Also you seem hell bent on equating mathematics with counting (quantity). Do you admit that other structures exist within the realm of mathematics which are unrelated to numbers?

Surely there is some use for things like groups, lattices, graphs and all those other fun things that aren't numbers.
 
  • #33
Hurkyl
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Sigh, just noticed my empty sets came out as the wrong character, and they are again. Bah, what's the code to get the pretty null set character?. :frown:


Using Cantor's notation for the Natural Numbers the set { {{Ø}}, Ø} would actually be undefined.

{ {{φ}}, φ} is indeed not an element of the typical set theoretical model of the natural numbers, but it is a perfectly well-defined set.


Cantor's set theory does not require any particular order to the elements in his definition of the Natural Numbers.

I'm not sure precisely what you mean, so I will cover this base. The term natural numbers implies some facts about the successor relation from which one can define an order.


From the purist philosophical notion it is impossible to have say, 2 basket balls, according to Cantor, but cause these objects are identical they can only be viewed as one object.

Recall that {a, b} has only one element if and only if a is equal to b; one can certainly use an equality relation for which two distinct basketballs really are inequal. For example, one could require that if two objects are basketballs, then they are equal iff they occupy the exact same points in space.


What are you suggesting here [about a 1-1 correspondence]?

I'm suggesting that the set {φ} is not used anywhere in the definition of a 1-1 correspondence. In fact, no mathematical object typically labelled '1' is required. A 1-1 function is simply an invertible function between sets.

Such objects, of course, may be used to describe a 1-1 correspondence, but are not required. E.G.

f is a 1-1 correspondence between A and B iff:

f is a subset of AxB (x = cross product)
For all a in A there exists b in B such that (a, b) is in f
For all b in B there exists a in A such that (a, b) is in f
For all x, y, z: (x, z) = (y, z) implies x = y
For all x, y, z: (x, y) = (x, z) implies y = z


The idea of One is defined in formal mathematics as the feeble quantitative idea of set containing the empty set.

The "feeble quantitative idea" of a set containing the empty set is a model, not a definition.


There is no other definition for One! If there was it would be an ambiguous concept.

Multiple definitions do not necessarily imply ambiguity; if all of the various definitions of "One" satisfy the same basic set of rules of manipulation (such as 1 + 1 = 2), then there is no ambiguity when using those rules of manipulation.


There are no other formally accepted definitions of One that I am aware of.

A typical formal definition of 1 is "the identity element of a multiplicative group".


Endless things don't have a size, they are simply endless.

"Endless" sets do have cardinality. Some cardinal numbers are finite, some are infinite. Some infinite cardinal numbers are bigger than others.

In the finite case, the notion of cardinality coincides with our intuitive notion of size. In the infinite case, cardinality retains some of the important properties of the finite case and can be used in the same way as the notion of size in proving theorems. Thus, in typical mathematical fashion, when trying to explain concepts in an intuitive way, we use the term "size" since it is so similar.


We can still address them using other formalisms of logic, but pushing this job onto mathematics only serves to further muddle the idea of quantity. And this is one of my biggest concerns.

It's not one I share. A rose by any other name still smells as sweet; the concepts, ideas, and definitions are all the same whether you call the formalism mathematical or not.


In my mathematical formalism...

In traditional set theory, a set is called infinite if and only if it is not finite. (A finite set is one that can be put into 1-1 correspondence with the set Nn; the set of natural numbers less than n). Sounds the same. :wink:

Actually, off the cuff, I can't remember anywhere in set theory that the term "infinity" appears; just the adjective "infinite", whose meaning is just what you describe in your formalism.

Of course, you hear it when people try to give nonrigorous intuitive explanations, such as one infinity being bigger than another infinity; the proper phrasing would be that one infinite cardinal number is bigger than another infinite cardinal number.


By the way, Cantor's diagonalization argument doesn't hold water. It's a complete sieve, and I reveal it's fallacy in my book.

[?]

Proof that for any set S, |S| < |P(S)|: (the cardinality of any set is less than the cardinality of its power set)

Let &pi; be any function from S to P(S).

Define the proposition Q(x) as, for any x in S:

Q(x) := x is an element of &pi;(x)

Define the set T by:

T := {x in S | ~Q(x)}

Note that T is a subset of S, therefore an element of P(S).

Now, suppose that &pi;(t) = T for some t in S.

If t is an element of T, then by the construction of T, ~Q(t) is true, which means that t is not an element of &pi;(t) = T, which is a contradiction.
If t is not an element of T, then by then construction of T, ~Q(t) is false, which means that t is an element of &pi;(t) = T, which is a contradiction.

Thus, the presumption that there is a t in S such that &pi;(t) = T is false. In particular, this means that there is an element T of P(S) such that &pi;(t) = T has no solution.

We have concluded that for any function &pi; from S to P(S), there is an element of P(S) that is not in the image of &pi;(S). This is just the definition of |S| < |P(S)|, so the theorem is proven.


Which step doesn't follow?
 
  • #34
NeutronStar
419
1
Joy Division wrote:

Just out of curiousity, are you working from Cantor's original work?

Yes.

Joy Division wrote:

There has been progress in set theory since then I wonder if you've looked into that?

I am aware that many band aids have been put onto Cantor's original work if this is what you mean. But as far as I'm concerned the idea of the empty set it the problem, so why keep putting band aids on it, why not just correct it?

Joy Division wrote:

What about Bertrand Russels category theory? That was one competing notion for the foundations of mathematics, it didn't work though and Cantor's set theory did *shrug*.

Category theories are fine for what they are. But they don't properly represent the idea of quantity.

Joy Division wrote:

Also you seem hell bent on equating mathematics with counting (quantity).

I absolutely am. For two reasons:

1. The interesting thing about the universe is that it displays a quantitative nature. It is the quantitative nature of the universe that has enabled use to define its physics . It's physical properties as we call them, and the quantitative relationships between these properties. We didn't invent that concept, we recognized it in the universe.

Kronecker actually had a point when he said, "God created the Integers, the rest is the work of man."

Kronecker did not accept Cantor's empty set theory either. (among many other mathematicians of that time)

2. I firmly believe that all of mathematics can be defined and comprehended using only a concept of quantity. Adding superfluous concepts servers what purpose? To confuse the idea of number?






Joy Division wrote:

Do you admit that other structures exist within the realm of mathematics which are unrelated to numbers?

I do admit that other forms of logic have been placed under the umbrella of mathematics . My question is simply this, "What makes them belong to mathematics?" Does poetry belong to mathematics?

I mean, what qualifies something as belonging to mathematics? This really was the question asked at the beginning of this thread, "What makes mathematics?"

As far as I'm concerned mathematics should be about the idea of number. Period amen.

If a logic isn't about the idea of number then it can still be a valid formalism in it's own right, but why call it mathematics? Where do we draw the line? Like I said, what prevents poetry from belonging to mathematics? What is the criteria for being part of mathematics? Is any form of logic now considered to be mathematics?

I personally don't consider Boolean Logic to be a part of mathematics, yet I use it all the time. It just has nothing to do with the idea of number that's all. Yet, it is often taught in mathematics departments at colleges, and it is often included in many mathematics textbooks. I think it is fine to include it in educational mathematics, but it should be made clear that it is not formally part of mathematics or the idea of number.

Now obviously binary arithmetic is mathematics, but that's not the same thing as Boolean logic. Binary numbers do reflect the idea of quantity.

Joy Division wrote:

Surely there is some use for things like groups, lattices, graphs and all those other fun things that aren't numbers.

I never said anywhere that there is no use for logical formalisms that don't rely on the idea of number. I merely said that they shouldn't be automatically tossed under the umbrella of mathematics. Mathematics should be about the idea of number.

Unfortunately I am not a mathematician. I am not up on groups. I was hoping to take that course this fall. I really should look into that concept. I'm not sure whether it is based on the idea of quantity or not. I do know that a lot of things that mathematics often think are not based on the idea of quantity actually are founded on that idea. Like probability theory for example.

As far as graphs are concerned I absolutely see them as being based on ideas of quantities. Graphs are nothing more than a visual representation of the relationships between quantities. And that's certainly fair game for mathematics. I see all of geometry as belonging to mathematics as well, because geometries are really nothing more than a quantitative representation of space and/or time as we perceive it. Or perhaps I should say, as the universe exhibits it.

I have a sneaky suspicion that group theory probably is drifting away from the idea of quantity, although I can't say for sure since I am not educated in group theory. But if it is, it only goes to prove my point that mathematics is drifting away from the idea of quantity.

My whole concern is that the universe has revealed to use it's quantitative nature, and that it is consistent in its behavior relative to this particular quality.

Now if there are other qualities that the universe has besides quantity that would be interesting to know. But at this point in time we are not aware of these qualities.

So if we have a formalism that does not represent the idea of quantity then what makes us think that we can use it to describe the behavior of the universe?

And even if one of those other logical system should happen to work. Wouldn't it be nice to know just what the hell it is describing? I mean, if it was just a haphazard offshoot of mathematics because we aren't paying attention to what we are modeling, then do we even have a clue what it is that we are modeling?

I think it would be nice to separate these formalisms so we know what the heck we are doing. Then if one formalisms starts popping up interesting answers we can turn to it and ask, "What quality is it describing?".

Right now we wouldn't do that because we are under the wrong impression that it is just some mysterious mathematics. Where mathematics no longer has anything to do with the original idea of quantity that we started out with.

Not to mention the fact, that if we go back and look at Cantor's set theory we will discover that (and this is a conditional statement):

IF,

Cantor's set theory is supposed to represent the quantitative property of the universe that we intuitively understand as number,

THEN,

Cantor's set theory is wrong!
 
  • #35
NeutronStar
419
1
On Radioactive Waves wrote:

We use theorems and postulates based on axioms. We use set theory. Are axioms based on set theory? or vice versa? More specify, what makes mathematics? What is it's primary basis?

It seems to me the entire basis of mathematics is the number 1, and its relation to itself.

I know one of you math whizzes should be able to explain this...

I thought I'd just quote the original post to this thread to remind readers of the original intent here.

Please feel free to jump in and respond directly to the original post at any time. :smile:

No need to address any current discussions. Unless ya wanna. :wink:

We are just voicing opinions and sharing food for thought. All food is welcome!
(I hope)
 

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