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What makes mathematics?

  1. Jul 19, 2003 #1
    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...
     
    Last edited: Jul 19, 2003
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  3. Jul 19, 2003 #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.
     
  4. Jul 19, 2003 #3
    Can you tell more about the "Zermelo-Frankel (ZF) axiom" ?
    Thanks.
     
  5. Jul 19, 2003 #4

    selfAdjoint

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    Well here is a link, Zermelo-Frankel axioms , 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.
     
  6. Jul 20, 2003 #5
    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/
     
  7. Jul 20, 2003 #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:

    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 existance 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 existance 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.


    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!


    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")


    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)


    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.
     
  8. Jul 21, 2003 #7
    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.

    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.

    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.
     
  9. Jul 21, 2003 #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.



    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.


    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.
     
  10. Jul 22, 2003 #9
    Wow, !

    Thanks Hurkyl (whose appearance doesn't suprise 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, theres 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)
     
  11. Jul 22, 2003 #10
    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.
     
  12. Jul 22, 2003 #11
    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 whats been stated by all means do jump in- I'm learnig just watching this discussion.
     
  13. Jul 23, 2003 #12
    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.

    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.

    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 in to 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?

    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:

    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:
     
  14. Jul 23, 2003 #13
    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?
     
  15. Jul 23, 2003 #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".
     
  16. Jul 23, 2003 #15
    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.

    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.

    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.
     
  17. Jul 24, 2003 #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.


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


    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.


    I don't have a problem with it.
     
  18. Jul 24, 2003 #17
    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 text books 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?
     
  19. Jul 25, 2003 #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.
     
  20. Jul 25, 2003 #19
    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.


    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?
     
  21. Jul 25, 2003 #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.


    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).
     
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