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arildno
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You should start with proving proof with no idea about what must be present in a valid proof. Or perhaps not..
matt grime said:Erm, no, that isn't quite what prove means in mathematics, is it? To prove something one needs a hypothesis from which to make a deduction.
Ok, so prove minus, then, prove 3, prove composition of functions.
arildno said:Nonsense!
You INTRODUCE the mathematical symbol "2" by defining
2=1+1
You've got some silly, unmathematical preconception about what "2" is; get rid of that.
Technically I disagree with that. Peano relies on the idea of 1 which has been formally accepted by the mathematical community to be defined by Cantor's definition. Therefore any reference to the number 1 is automatically a reference to the empty set by default. (In other words, Peano doesn't actually define the number 1 in his axioms, he merely uses the preexisting concept)jcsd said:We don't have to start out with the empty set in order to build Peanos axioms
Now this has always been of interest to me. Since 1 = {0} = {{}}jcsd said:0 = {}
0* = 1 = {0} = {{}}
(0*)* = 1* = 2 = {0,1} = {{{}},{}}
matt grime said:The OP was aksed to "prove red" and he didn't (though he appears to think he did). He gave a "definition" of red. That in itself doesn't "prove" red.
drcrabs said:So basically after reading these pages, you are trying tell me that you actually can't prove addition?
StatusX said:also, all this talk about set theory is, in my opinion, off topic. set theory is not the only way to define addition, and it implies there is only one way it could work. its possible to imagine a universe where 1+3 is not equal to 2+2. addition is, whether you like it or not, an empirical law, from which its set-theoretic defintion was derived.
jcsd said:Anyone who starts to talk about 'empirical laws' when talking about math proofs is most defintely on the wrong track. Maths does not claim to describe reality, the axioms of mathethamtical systems do not come from the observation of reality, they are true for the system because we define them to be true. It is pewrfectly possible to have a statement that is axiomatic in one mathematical syetm but false in another.
Actually I wish that was true.Hurkyl said:Science/philosophy decides what axioms you should be using, and mathematics tells you what those axioms imply.
drcrabs said:Yea waddup Grimey. I've noticed that you realized that i gave a definition. The purpose of the 'definition of red post' was to point out that the person who was asking me to prove red was getting off the topic
NeutronStar said:Actually I wish that was true.
NeutronStar said:IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed".
CrankFan said:You've not given any justification for your belief that the foundation of mathematics, as it's currently understood, is insufficient or inconsistent. Instead, you've insisted that some facet of the current approach is wrong (like say, defining 0 first) and fail to show how it leads to either a contradiction or limitation in characterizing some useful area of mathematics. Essentially, you don't like construction of N starting from 0, but can't articulate why this is problematic; it's just a (religious) belief.
NeutronStar said:I think that there's been a general consensus among the responders to this thread that addition cannot be proven (or as some claim it is even meaningless to ask such a question).
NeutronStar said:I was merely pointing out that the particular axiomatic framework that is preventing the question from being meaningful was introduced rather recently (with respect to the entire history of mathematics). And that the theory that prevents it is Georg Cantor's empty set theory.
NeutronStar said:I believe that everything that I've said here is true. I believe this on solid logical grounds.
NeutronStar said:The mathematical community as a whole is well aware that there are logical problems associated with set theory.
NeutronStar said:Originally (that is to say that before the time of Georg Cantor) a set was intuitively understood as a collection of things. Cantor then introduced the idea of an empty set. A set containing nothing.
NeutronStar said:Well, this is actually a logical contradiction right here.
NeutronStar said:We have a collection of things that is not the collection of a thing. This is a logical contradiction to the very definition of what it means to be a set.
NeutronStar said:Similar objections actually came up by other mathematicians at the time that Cantor proposed his empty set idea. These objections were never (and I mean NEVER) fully resolved.
NeutronStar said:However, later on (possibly even after Cantor's death, I'm really not sure on that) the mathematical community started seeing other problems cropping up. For example, there must be a distinction between a set, and an element within a set. If this distinction isn't made then 1 would equal 0.
NeutronStar said:In other words, by Cantor's empty set theory
0 = {}
1={{}}
Now if there is no distinction between an element, and a set containing a single element then there is no difference between Cantor's definition of zero and One.
NeutronStar said:The bottom line to all of this is that Cantor's empty set theory is logically inconsistent.
NeutronStar said:So big deal you might say. It's a trivial thing just let it go and get on with using the axioms.
I'm not actually having any misconceptions between naive set theory and ZF. In fact, I see them as totally separate things.jcsd said:I think Neutron Star one of your main misconceptions is the between naive set theory and ZF set theory.
Hurkyl said:NeutronStar, I'll start with a simple question: do you reject the number zero?
NeutronStar said:The whole reason why Cantor introduced the idea of an empty set in the first place was to avoid having to tie the concept of number to the concept of the things that are being quantified. In other words, at the that time in history it was extremely important to the mathematical community that numbers be "pure". I might add that this was much more of a religious notion than a logical one so if we want to claim that someone was being religious rather than logical we should look at the mathematical community around the time that Georg Cantor lived!
Abel said:There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has lead to so few of the so-called paradoxes.
Hurkyl said:As you can see, it was a practical necessity that led to the search for a "pure" way to define the numbers, so that results could be rigorously justified, in the spirit of Euclid.
DO this first.drcrabs said:I reckon addition is a basic skill. But can it be proved?
How?
The whole idea for me is that it works, and it works in a fully comprehensible fashion that leaves nothing undefined.
Moreover, it's actually a more accurate description of the true quantitative nature of our universe.