Is Addition Really a Basic Skill?

[Hurkyl]

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

Well, you already reject the notion that you can put irrational notions into a set. If you also reject the notion that you can put solutions to polynomials into a set, then that doesn't leave any difference between the algebraics and solutions to polynomials, does it?


By the way...

Would you agree there is a set of all polynomials with integer coefficients? There are at least two commonly used ways to represent a polynomial: as a function and as a list of coefficients (which is simply a finite list of integers).

If you agree there is a set of all polynomials, you will probably agree that there is a set of all pairs of the form (p, n) where p is a polynomial and n is a natural number.

(A subset of) this last set is a representation of all solutions of polynomials with integer coefficients: each solution is represented as a polynomial that it satisfies and an integer denoting which solution it is.


This is actually a sort of ad-hoc construction that is based on the familiar structure of the rational and algebraic numbers. There's a more generic construction used in field theory to do the same thing. However, like this one, it starts from the fact that there is a set of polynomials over a set of variables.

Irrational numbers don't have a property of individuality

I've been trying to avoid this sort of confrontational response, but I don't think I can this time -- this sentence is entirely meaningless until you can convey what you mean by "individuality".

You don't happen to have a URL on the Internet that explains this problem.

Unfortunately, no. This sort of problem tends to come up in measure theory, as the proof that the rational numbers have measure zero in the real numbers.

But the essential points of it are this:


(1) Every point is contained in a "neighborhood" of length less than 1/2^n for any positive integer n.

The rational line has this property. For any point x, one example is the interval (x - 1/2^(n+1), x + 1/2^(n+1)).

(2) There is a bijection from the points to the natural numbers.

(3) The "total length" of a collection of neighborhoods is no greater than the sum of their individual lengths.

 

[Hurkyl]

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

(emphasis mine)

Bleh, I missed this the first time through. The italicized statement is exactly why you're simply arguing semantics. Well, half your argument, anyways.

It seems your only problems with formal set theory are:
(1) It's "incomprehensible".
(2) It calls its objects of study "sets".
(And these two points are interrelated)

When one of the primary thrusts of your argument is that you don't like the choice of words, then yes, you are arguing semantics.



Anyways, this morning I was reading stuff on nonstandard analysis, and it reminded me that I wanted to make a point about comprehension. (the mathematical meaning of the word)



The way I see it, set theory has always been about modelling concepts and ideas... well, more precisely, modelling logic.

Sets were supposed to be the answers to questions. If I have some logical predicate P, then I can talk about the set of all things satisfying that predicate.

For instance, P(x) might mean "x is an even integer", so I can talk about the collection of even integers. This is part of where we diverge semantically, because I've never found it unusual to refer to the objects that satisfy a predicate as a "collection".

Anyways, when Cantor developed his naive set theory, this was one of the natural ways to build sets: given any logical predicate P, we have the set

{x | P(x) is true}

This is called unrestricted comprehension, and this axiom was the problem with Cantor's set theory. In ZF, it was replaced with the axiom of subsets: given any set S, we can select out the objects of S that satisfy the predicate P. That is, the following is a set:

{x in S | P(x) is true)



Anyways, stepping forward a bit, this idea of modelling logic itself is the impetus for model theory, one of the more powerful branches of formal set theory. I don't know too much about it, so I'm probably understating its power here, but it has this interesting property:

If I have any consistent theory written in first-order logic, then that theory can be recast into set theory: there exists a set that acts as the "set of objects" for that theory, and there exists (set-theoretic) relations that behave like the relations of the theory, and all of the axioms of the theory hold.


One example of a model is Cantor's natural numbers that model Peano's axioms. The objects are the familiar:
{}, {{}}, {{}, {{}}}, ...
There is only one relation in Peano's axioms: the successor relation "x is the successor of y". This is modeled as the relation:

x is the successor of y iff x = y U {y}

and one can go on to show that all of Peano's axioms hold for Cantor's natural numbers.


If you're taking the Peano axioms as the definition of natural numbers, then Cantor's model is just as good as any other interpretation of the theory.

 

[NeutronStar]

Well, you already reject the notion that you can put irrational notions into a set. If you also reject the notion that you can put solutions to polynomials into a set, then that doesn't leave any difference between the algebraics and solutions to polynomials, does it?

You've simply misunderstood what I am trying to say (or I didn't state it very well,… whatever).

I'm only saying that in my formalism you can't technically call "solutions sets" that contain irrational numbers mathematical "sets" by the definition of a set in my formalism because my formalism does not allow the concept of an irrational number to be an element in a set.

If the "solution set" of a polynomial does not contain any irrational "numbers", then it can qualify as a mathematical "set".

It's really no different than the idea of a function that I referred to before. Mathematics currently recognizes relationships that don't quality as functions and therefore don't satisfy the definition, and/or rules for functions. It's really the same thing. I'm just saying that any collection of objects that contain irrational numbers simply doesn't satisfy the definition of a set in my formalism and therefore it would be improper to call it a set, or expect it to "behave" like a set.

There's only one notion here. Irrational concepts do not fulfill the requirement of individuality in my formalism therefore they cannot be thought of as individual elements in a set. They simply don't satisfy the definition of individuality in my formalism. Much like some relationships in mathematics don't satisfy the definition of a function and therefore can't be called functions because they don't behave like functions.

I never meant to imply that solutions to a polynomial can't be thought of as a set. It's only if that solution contains irrational numbers is when it can't be called a set.

I've been trying to avoid this sort of confrontational response, but I don't think I can this time -- this sentence is entirely meaningless until you can convey what you mean by "individuality".

I agree. It is paramount to my formalism that I have a clear and workable definition for the conceptual "meaning" of individuality, and I do.

However, because this is the foundation of my formalism, it is imperative that it be introduced properly. Unfortunately Internet message boards aren't the best media to convey that information. This is especially true when speaking to mathematicians who are fixated on axiomatic systems. It would probably be a little easier to describe this to a philosophical audience that is more open to conceptual ideas.

In short, in the crudest sense an object dose not have a property of individuality unless you can prove that you can collect it in its entirety. I hold that it is impossible to collect the concept of an irrational number in its entirety.

There are so many really eloquent ways that this can be stated in a live lecture. And because its such a new concept to mathematicians it really pays to go through them all in detail with visual aids.

In addition to this, I wouldn't even give this lecture as a stand-alone lecture. It should be followed by a prerequisite lecture on why the property of individuality is so important to the concept of number. That lecture is a flash-back to kindergarten to take a loot once again at our original intuitive comprehension of the idea of number. We exam flash cards that represent the various natural numbers. Then I introduce a flash card that contains the irrational concept and ask everyone to tell me what quantity in represents. No one can say. Not because they aren't smart enough, but because it obviously impossible!

This clearly shows visually and conceptually to why irrational concepts can't be thought of as quantities in the normal intuitive way. In other words, it clearly shows that they are a concept other than quantity. I actually show much later exactly what this other concept is.

In any case, then we move onto lecture two where the formal definition of individuality is introduced. I'm not sure that it can be reduced to something as simple as "There exists an individual thing". Kind of like Cantor's original axiom, "There exists an empty set". But I do offer a conceptual definition that can be clearly understood intuitively both concretely and abstractly.

After that, I show how this new concept of a set actually makes more sense conceptually to explain the arithmetic operations, the concept of negative numbers (and the concept of imaginary numbers). It also clearly shows the two faces of addition and brings home the reason why division by zero (which is not a number in my formalism) is ill-defined.

After that I like to show how this change does not affect the calculus at all. That's also a good time to give the proof why a finite line segment can only contain an finite number of points and therefore that the idea of anything possessing a property of quantity (by this new definition of number) is necessarily quantized. (i.e. concluding from pure logic that the universe necessarily must be quantized).

Well, it's not exactly from pure logic now is it? I mean, I've been claiming all along that my fundamental definition of number and the recognition of this property of individuality stems from the fact that this is a correct reflection of the observed property of the universe that we call "quantity" or "number". So my formalism isn't based on pure thought along, it actually has it foundation in observation and experiment. Wow, my mathematics really can be called a "science" because I used the scientific method to construct it! Current mathematical formalism really has no right to claim that it is a "science" because it does not employ the scientific method.

In any case, there's room for even more lectures on the topic. Looking at the flaws in the logic of Cantor's diagonal proof would be good to go into. A close examination of precisely what's going on there actually supports my new mathematical model of quantity and shows how this property of irrationality has been confused by cantor as a property of quantity, or as he calls it, "cardinality". That's just so totally incorrect.

Alright, in layman's terms for a thing to be considered to have the property of individuality you have to be able to prove that you can collect it in its entirety. If you believe that you can collect an irrational concept in its entirety see Cantor's diagonal proof! He actually proved my point! Therefore I claim that irrational numbers do not have a property of individuality. Therefore they cannot be considered to be elements in quantitative sets.

Actually it's not just irrational numbers. Any object that cannot be shown to possesses a property of individuality. In other words, if you can't prove that you can collect it in its entirety, then you can't say that its "one thing" from a quantitative point of view. In other words, you can't claim that you can count it as a "single element". The whole idea of bijection or a one-to-one correspondence is totally meaningless if the things that you are putting into a one-to-one correspondence don't have the property of being "One" thing.

I really need to stop wasting my time here because I'm sure that this isn't going to do anything but generate more question.

By the way, Cantor was actually aware of the need to define the property of "individuality". He did so in a non-quantitative way that the thought was quantitative. He simply introduced the concept of an empty set. His idea is undeniably a "unique" thing. He thought that because it can be readily seen to be a "unique" idea he could also assume that it is a "single" idea, thus it must be quantitatively "One" idea. Therefore it properly represents the quantitative idea of "One". Thus it can be treated as a single element in the set containing the empty set. (i.e. the definition of the number One).

But where he went wrong was that his empty set isn't actually quantitatively one thing. You can't show that you have collected it in its entirety. In other words, where does a collection of nothing begin or end? There's no way to know if you have collected it in its entirety. The concept of the empty set does not have a quantitative property of individuality. It merely has a qualitative property of being a unique idea. That's totally useless for defining the concept of quantity. So Cantor truly did blow it.

I give him credit for being a genius non the less though. I mean the fact that he could take an incorrect concept and run with it so far, and sell it to the mathematical community says something about his genius. But he's qualitative description of the idea of number is incorrect when compared to the actual property of quantity that is exhibited by the universe. This is why I continue to toss out my conditional statement,…

IF "mathematics is supposed to correctly model the property of the universe that we call quantity" THEN "current mathematical formalism is incorrect", or more precisely, the concept of an empty set is incorrect. Because after all, calculus is still valid!

Unfortunately, no. This sort of problem tends to come up in measure theory, as the proof that the rational numbers have measure zero in the real numbers.

But the essential points of it are this:


(1) Every point is contained in a "neighborhood" of length less than 1/2^n for any positive integer n.

The rational line has this property. For any point x, one example is the interval (x - 1/2^(n+1), x + 1/2^(n+1)).

(2) There is a bijection from the points to the natural numbers.

(3) The "total length" of a collection of neighborhoods is no greater than the sum of their individual lengths.

Well, I'm still interested in this, but I really don't have time to think about it just now.

 

[Hurkyl]

It would probably be a little easier to describe this to a philosophical audience that is more open to conceptual ideas.

I would argue that this characterization is because people with new theories don't tend to say "Hey, look at my theory!": they instead say "Hey, your theory's wrong because it's not my theory!" much like you have.

I hold that it is impossible to collect the concept of an irrational number in its entirety.

Let's pin this down... I think, as an example, you would say that:

"The concept of \sqrt{2} cannot be collected in its entirety".


What's wrong with "\sqrt{2} is the positive solution to x^2=2" or "It's the length of the hypotenuse of a triangle with two sides of length 1"?


Someone once gave an example of a dog given the name "43", and went on to explain how people often confused the string of symbols "43" with the number it often represents.

I hope you're not doing the same thing with irrational numbers -- confusing an irrational number with its decimal representation.


------------------------------------------------------------------

Now, let's look at the next couple paragraphs. You apparently realize that the statement:

"irrational 'numbers' can represent a 'quantity'"

is going to be a point of contention. (why else would you devote a whole lecture to it?)

So if you know people are going to disagree, why do you attempt to make your point by simply saying it's "obvious" that you're right?

Even worse, you build upon this point. You've not only not convinced anyone of this point, but you've given them reason to think you cannot support this point, so there's no way they're going to accept anything built upon this point...



Now, you could have just said that irrational numbers don't satisfy whatever concept and left it at that...

--------------------------------------------------------------

 

[NeutronStar]

I hope you're not doing the same thing with irrational numbers -- confusing an irrational number with its decimal representation.

Scream !,...

Who's doing this ?,...

This is precisely what Georg Cantor is doing with his diagonal proof!

Actually, I'm all for allowing concepts called "irrational numbers", but only if they are recognized as being rounded off at some point so that they can be finite concepts. Although it is also very useful to understand why they can't be rounded off. (that may sound paradoxical but it really wasn't actually)

Actually if we allow irrational numbers to be represented by finite symbols like say \pi, \sqrt{2} etc. then Georg Cantor's diagonal proof falls flat on its face!

After all we can simply name the symbols S_{(n)} where n is a natural number subscript for each irrational number. We can create an endless quantity of these symbols so we don' t need to worry about running out of them.

However, once we have done that then we can easily put the real numbers in a one-to-one correspondence with the natural numbers because the symbols that we use to represent them are subscripted by the natural numbers.

Therefore |R| = |N| in that case!

Georg Cantor's diagonalization proof rest entirely on the idea of treating the irrational numbers as decimal representations!

So don't look at me! I'm totally against that idea altogether.

So the question now becomes this,... does the concept of an irrational number have a property of individuality outside of its decimal notation.

My answer is a resounding NO!

Why not? Well, now we need to move into the area of self-reference and theories like Kurt Gödel's inconsistency theory for that. But I'm not sure that I really want to get that deep into things here on an Internet forum board.

 

[NeutronStar]

Now, you could have just said that irrational numbers don't satisfy whatever concept and left it at that...

Ok, irrational numbers don't satisfy the condition of being individual things because they aren't individual things.

All irrational numbers are a result of self-referencing situations. Thus they are relative concepts referencing relationships between interconnected (self-referenced) concepts rather than being individual quantities in their own right.

They arise from self-referenced reflections. Like looking in a mirror. They are illusions. They don't exist outside of their self-referenced situations.

Does that make it clear?

Probably not.

Actually the deal with the irrational numbers is really just an aside to the bigger picture. This is only one of many enlightening concepts that fall out of correcting the definition of the number One as it is based on set theory.

 

[jcsd]

Sigh I wasn't going to argue, but Neutron Star you really do need to realiz ethat you much of these disagreemnts are the result of your own misconceptions and not only that, but your not going to persuade anyone otherwise because this fact is obvious to all!

George Cantor considered a representation of real numbers to prove a general property of the real numbers, nowt wrong with that. But you are cofusing the represntation with the actual mathematical object. Where exactly do we need to round THIS represnation of THIS irrational number: \sqrt{2}? (and note that this representation is a much more useful one than 1.41421...)

 

[NeutronStar]

Sigh I wasn't going to argue, but Neutron Star you really do need to realiz ethat you much of these disagreemnts are the result of your own misconceptions and not only that, but your not going to persuade anyone otherwise because this fact is obvious to all!

Well, just for the record, I didn't start a thread of my own to convince anyone of anything.

I just keep coming back to this thread and responding to questions because I have no real discipline. (ha ha)

But you're right this really is pretty fruitless on my part. Besides, even if I actually got through to someone what good would it do me? They would probably just run off and take credit for the idea anyhow and no one would ever know who NeutronStar even was!

There's really no point in wasting my time here further. In fact, I retract everything I said. I was grossly wrong about everything because I'm a stupid uneducated moron. I see that now. Thanks for pointing it out.

 

[jcsd]

I thought you would retract evrything you said! :)

But seriuosly my point is that I am understanding everything you are saying, but I also undretsand why what you are saying is wrong, infact many of your observations are not even new, we're going over some very well-tordden territory here.

 

[Hurkyl]

Who's doing this ?,...

I'm pretty sure you mentioned earlier about truncating decimal representations of irrational numbers, and you did again in this post, so I hope you can understand why I get the impression that nonrepeating decimal expansions are a significant part of your problems with irrational numbers.


Cantor doesn't assume that real numbers are decimal expansions: just that there is a bijection between real numbers and decimal expansions.


Cantor's argument can even be written purely analytically:

Suppose f is any function from the positive integers to the real numbers.

Then, compute the following:

<br /> \phi := \sum_{n=1}^{\infty} 10^{-n} \left( 9 - \lfloor 10^n f(n) \rfloor + 10 \lfloor 10^{n-1} f(n)} \rfloor \right)<br />

Then you can show that f(n) \neq \phi for any positive integer n.

 

[NeutronStar]

Cantor doesn't assume that real numbers are decimal expansions: just that there is a bijection between real numbers and decimal expansions.

Well that's an assumption that I'm not prepared to make. How does Cantor justify such an assumption?

This goes back to the dog name "43". Just because we can create a string of numerals doesn't mean that we have created a valid and meaningful concept of number. That's simply invalid logic.

This is especially true when those strings are endless decimal expansions. Cantor's assumption that there exists a bijection between the reals and all possible decimal expansions doesn't hold water for me.

Look at it this way.

All irrational numbers can be represented by non-repeating decimal expansions.

That's true!

But to then conclude that all non-repeating decimal expansions represent irrational numbers is incorrect logic! It's simply wrong. Any logic instructor will quickly tell a first-year college student that such a backward conclusion is illogical and doesn't hold water.

Here's another example.

I claim that all irrational numbers are a result of a self-referenced situations.

While I believe that this is true the opposite is not true.

In other words, all self-referenced situations do not give rise to irrational numbers!

Cantor is assuming (with very incorrect logic) that all non-repeating decimal expansions qualify as irrational numbers. But he has absolutely no logical basis to make that claim. It simply isn't logically sound reasoning. Therefore I deny the bijection assumption upon which his summation equation rests.

Sorry. I'm not trying to be difficult. I'm just being logical.

 

[arildno]

Cantor's diagonal argument constructs an increasing, bounded sequence of rationals whose limit cannot be on his list.

But from what I know, we partition the set of ALL such sequences into equivalence classes, which are then identified with the reals.
Hence, what Cantor constructs, is a REAL number.

I might be wrong on this, and deserve a scathing from several sides..

 

[matt grime]

So far we have ascertained that the only collections of objects you will allow to be called sets are collections of (possibly infinitely many) natural numbers.

This of course begs the question of why you accept the notion of a bijection from N to R since R isn't a set, and functions are defined for sets (usually).

The Real numbers are equivalent to the set of cauchy equivlance classes of convergent strings of rational numbers, or the dedekind cuts (are rationals acceptable, why?), and thus given any element , x, of this complete metric space, it is easy to construct a decimal expansion, which modulo recurring 9s, which we will formally declare not to happen, ie replace them with 0s instead. It is then rather easy to construct bijections between R and and P(N) the power set of the natural numbers. Is the power set operation allowed in your theory? Ie is P(N) a set? Anyway, given P(N) we can apply the theorem that no set is bijective with its power set (a result that is true for all sets in the model of ZF wherein we are operating). So what's wrong here?


What on Earth is a self referenced situation?


You also appear to be arguing that the only sets with quantity are finite ones. I doubt that many here would disagree, per se, and that all other sets are infinite and that there is no quantitative difference between them. Well, that is then up to you to state what you mean by quantitative for infinite sets.

I agree that A implies B does not imply B implies A, your irrational/repeating decimal bit, but since we have a theorem that states all rationals have an eventually repeating decimal expansion (I allow of 0 recurring here rather than saying terminating), actually A iff B holds.

Note, though, that at no point in Cantor's argument does it say that any of the numbers on the list are or aren't rational, nor that the decimal so constructed is or isn't rational. Merely that given any countable list of real numbers there is a real number not on that list.

And, since a decimal represents a series, which has convergent partial sums that form a cauchy sequence of rationals, then it is a real number.

I'm sorry that you don't accept that, but your acceptance (or even understanding) is not required for it.

 

[Hurkyl]

Actually, my analytic version of Cantor's argument is slightly off -- I tried a bad shortcut. Here's a valid version:

Let g be the function on the set {0, 1, 2, ..., 9} where g(0) = 1 and g(n) = 0 for n > 0.

Let f be any function from the positive integers to the reals. Compute

<br /> \phi_f := \sum_{n=1}^{\infty} 10^{-n} g(\lfloor 10^n f(n) \rfloor - 10 \lfloor 10^{n-1} f(n) \rfloor )<br />

Then \phi_f \neq f(n) for all positive integers n.

 

[NeutronStar]

On Self-reference and Irrational Numbers

What on Earth is a self referenced situation?

Well let's see,…

Pi is the ratio of two properties of the same geometric object. Changing one of those properties coincidentally changes the other. In other words, you can't adjust the diameter of a circle without directly affecting the length of its circumference and vice versa.

The irrational number Pi arises from a self-referenced situation much like putting two mirrors back to back. Trying to adjust the image in one mirror has an instantaneous affect on the image in the other mirror.

The \sqrt2 is the number than when collected together the same number of times that it represents adds up to 2.

Think of it in terms of collections of things. Let q=\sqrt2 then q is the number that must be collected together the same number of times that it represents. In other words if we let q=2 then we must collect 2 together twice. But that equals 4 so that won't work. So let q=1. Then we must collect 1 together once. But that only equals 1 so that won't work.

So q must be somewhere between 1 and 2. Let's try 1.5 Well, that means that we have to collect 1.5 together 1.5 times. So collecting it together once gives us 1.5. Then collecting .5 more of it we get 2.25. Well, we're getting closer but we're still not right on the money, we're a little bit over.

As we continue this process we find that if we let q=1.4 then collecting that together 1.4 times gives us 1.96. Hey! That's pretty darn close to 2. In fact, if we could collect 1.4 together exactly 10/7 times we'd have it exactly! But wait a minute!

We can't do that because it's a self-referenced situation! Remember that q represents BOTH the number that we are collecting together AND the number of times that we must collect it. So q can't be both 1.4 and 10/7 simultaneously. If we change the number of times we collect q together, then we have also changed the value of the quantity that we are collecting.

So we're stuck in this self-referenced situation just like the back-to-back mirrors. Adjust one mirror and it instantly changes the image in the other mirror. The square root of 2 is a self-referenced situation. In fact, the square root of all numbers are self-referenced situations. Obviously all self-referenced situations don't result in irrational numbers. The square root of 4 for example is simply 2.

Let's try one more,…

Euler's number e is defined as e=\lim_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n

Here we can see that n is being self-referenced as it is in an expression that is being taken to the power of n itself. There are other ways to define e but I think that this definition clearly shows the self-referencing situation that's involved.

I have asked many mathematicians over the course of my life to present me with a "meaningful" irrational number that is not associated with a self-referenced situation. So far no one has been able to come up with an example.

The closest thing that they can offer are arbitrary strings of meaningless numerals. When they give me such meaningless strings of numerals I feed them to my dog named "43".

Although, I have yet to even see an arbitrary string of non-repeating numerals that isn't self-referenced in some fashion. Take the meaningless so-called irrational number 1.12123123423451234561234567 and so on. Just for clarity that's just 1, then .12, then 123, then 1234 and so on to create a non-repeating endless decimal. Then using Cantor's backward logic and claim that this string of numerals represents a valid concept of number.

Ok, just for fun let's say that we accept this as a valid number. Then I still claim that it is self-referenced because as the sequence grows it's dependent upon what came before it. In other words, each new additional 123,.. must be different from the previous 123,…. otherwise we'd have no way of proving that it never repeats. So self-reference is crucial to its very definition of being non-repeating. Therefore the property of self-reference is mandatory in this situation.

However, even if someone could give me a string of numerals that does not repeat and is not self-referenced I would still be skeptical that it satisfies the definition of a number. It just looks like dog food to me.

What I would really like to see is a "meaningful" irrational number (like pi or e) that is clearly not the result of a self-referenced situation. Then I could know that I'm wrong and retract my conjecture that all irrational numbers are the result of a self-referenced situation. Thus far in my life I have yet to see any such example.

 

[matt grime]

So you're going to ignore the maths and simlpy say that "because I can only think of one way of describing it, then that's it"? Well, as sqrt(2) isn't the number which when gathered together "itself" number of times we get 2, then we're ok, I suppose.

Why on Earth do you not accept decimal expansions? Ok, they aren't perfect (diadic representations), but they're useful.

Of course, and I'm willing to stake my house on this, you don't actaully know what the real numbers are do you? I don't mean that in any "gee, look what the stupid kid thinks" way, but as a common observation that, as we don't actually live in a platonic realm, then the real numbers need some rigorous foundation, one that isn't taught to the vast majority of people. One of which is as infinite strings of decimals.

I can give a non-self referential proof, perhaps, of sqrt(2), but as your idea of self referential is hand-wavy and vague, who knows.

Let's try anyway. Sqrt(2) is the dedekind cut given by in informal presentation, sup{x in Q : x^2<2}, or by the cauchy sequence of finite sums given by x_n is the maximal element in the set of rationals with no more than n non-zero terms whose square is less than 2.

Just because you interpret multiplication in that way of yours neither means that that is what it is, nor that even doing so is the only way.

The real numbers are "the" complete totally ordered field.

Of course, since you think "gathered together sqrt(2) many times" is how we define sqrt(2), then we could be in for an uphill battle. Moreover, your argument presumes that sqrt(2) exists and must in some way represent some quantity of which we can say: gathe something together sqrt(2) times, where as really it is a construction derived from the rationals, themselves derived from the integers, in turn derived from the naturals, which we'll presume exist, though they can be put on a rigorous footing if need be.

Also, e isn't usually defined to be the limit you give, and even if it were, then all rational (non-integral) numbers are defined in a self referential way: 1/n is the number that when gathered together n times gives 1, is a self referential formula, by which you appear to mean "has some symbol appearing twice", that defines 1/n. Oh no, better reject rationals. Actually, you really had better reject them, hadn't you... not to mention integers: surely k is the thing which when split k ways equally gives 1 to each, k referring to itself... but wait, what's 1?

I particularl like your observation that there are other ways to define e, but you don't list them, instead you only give one that supports your claim, when the claim surely is "for all", and proof by example will never do there.


e is sum over n in N of 1/n!

is that self referential since n appears twice even though it isn't a variable (it is a dummy variable, and index, is that ok?)?

Ok, what about e is the unique value of f(1) where f is the unique solution to f'=f with boundary condition f(0)=1?

Your particularly bizarre argument involving 10/7 and 1.4 is very strange indeed. Who said that collecting 1.4 10/7 times ought to imply that they are equal and that they are the sqaure root of 2? Apart from you I mean.

 

[NeutronStar]

Your particularly bizarre argument involving 10/7 and 1.4 is very strange indeed. Who said that collecting 1.4 10/7 times ought to imply that they are equal and that they are the sqaure root of 2? Apart from you I mean.

We obviously have some miscommunication here because I never even implied that collecting 1.4 10/7 times ought to imply that they are equal and that they are the square root of 2. In fact, I thought that I clearly stated that they can't be the square root of two because q can't be two different numbers simultaneously! That was my whole point. The number q represents a single number that must represent BOTH the number we are collecting together, and the number of times that we are collecting it. My whole point is that it's a self-referenced situation.

Your assertion that I believe that collecting 1.4 together 10/7 times ought to imply that these numbers are equal and that they are the square root of 2 is simply not what I said.

As far as I can see the whole idea of collections and their behaviors was the initial concept that sparked the whole idea of set theory. If the theory has be distorted so terribly that it can't even support this fundamental basic concept then that's really scary! Things are worse than I thought!

 

[jcsd]

The thing is though sqrt(2) clearly is a real number by the defintion of a real number (there are sveral equivalent defintions, but however which way you choose to play it that sqrt(2) is a realmnumber is unescapable!). Your working from some vague preconceived notion of a real number that you have inside your own head, but mathematicians already have a rigorous definition of a real number and if your preconceived notion does not fit this definition then you are not talking about rela numbers in the commonly understood sense.

 

[learningphysics]

Definition of addition and proof of basic laws

Hi Drcrabs. I saw your question, and I think I understood what you were looking for. Have a look at this site:

http://ndp.jct.ac.il/tutorials/Discrete/node50.html

They first define addition, and then prove the commutative and associative laws of addition (basic laws that we use day to day). It involves a lot of proof by induction.

I asked myself similar questions a few years back. How do I know the commutative law is true etc. Anyway I hope this helps.

 

[Hurkyl]

I'm sure you're familiar with the construction of the complex numbers from the reals, right? You can consider the complexes as ordered pairs (x, y) (i.e. the complex plane), and define addition and multiplication for them, etc.


You can do an entirely analogous thing for the square root of 2.


I'm going to define a number field (I don't know why it has that name, but it does) that is often named \mathbb{Q}[\sqrt{2}].

Take all ordered pairs of the form (p, q) where p and q are rational numbers. Here are the definitions for arithmetic operations:

(a, b) + (c, d) = (a + c, b + d)
(a, b) - (c, d) = (a - c, b - d)
(a, b) * (c, d) = (ac + 2bd, ad + bc)
<br /> \frac{(a, b)}{(c, d)} = \left( \frac{ac - 2bd}{c^2-2d^2}, \frac{bc - ad}{c^2 - 2d^2} \right)<br />

I can define an ordering relation, <, but it's a little more complicated requiring several cases.


We have the embedding of the rationals in this number field via:

a --> (a, 0)

(just like for complex numbers)

And it's fairly straightforward to check that:

(0, 1) * (0, 1) = (2, 0) = 2.

So, (0, 1) is a square root of 2, with the other being (0, -1).


And, just like we usually write the complex number (a, b) as a + bi, we would usually write the element (a, b) of this number field as a + b &radic;2

 

[drcrabs]

Hi Drcrabs. I saw your question, and I think I understood what you were looking for. Have a look at this site:

http://ndp.jct.ac.il/tutorials/Discrete/node50.html

They first define addition, and then prove the commutative and associative laws of addition (basic laws that we use day to day). It involves a lot of proof by induction.

I asked myself similar questions a few years back. How do I know the commutative law is true etc. Anyway I hope this helps.

Cheers bro

 

[matt grime]

It should also be said that in Hurkyl's field the polynomial x^2-2 splits, and hence indeed there is such a thing as the square root of 2.

As for the 10/7 1.4 thing, you implication was that you ought stop after two steps in the construction of the decimal expansion, and that because these two things aren't equal the square root of 2 doesn't exist in the real numbers. That is what is odd about your argument. I didn't say you claimed they were equal I claimed you implied that they *ought* to be equal, and that because they aren't sqrt(2) is self referenced (whatever that may actually mean) and therefore isn't a valid concept.



Are you going to answer the questions about self referencing and its implications?

note there is a real number whose square is 2 since R is complete and f(x)=x^2 is continuous f(0)=0, f(2)=4, thus the intermediate value theorem applies.

 

[Hurkyl]

I remembered some of the issues I remember reading about trying to do physics on discrete space-times.

The first issue is that you cannot have nice, neat, orderly lines. In order to approximate the observed Lorentz invariance and local isotropy, the points have to be scattered about in a chaotic fashion.

The second issue, which I understand is considered a killing blow, is that all known theories of discrete space-times predict that light from distant sources should be blurry, but not even the slightest blurring is observed.

 

[balkan]

ridiiiiicuuloouuuuussssss![/size]

 

[robert Ihnot]

I asked myself similar questions a few years back. How do I know the commutative law is true etc. Anyway I hope this helps.

To my mind, if you go to the supermarket and pay for two pops and three cans of tuna, the price is not changed if you pay for the tuna first and then the pop, or for that matter if you mix the items up at check out.

Now if things were not like that, then if a primitative man put three arrow heads and two pieces of wood in a hiding place, the material could change if he did not extract them in the same order.

If during roll call, they started out by lining up from Z to A rather than from A to Z, would the number of members change?

So how could any other system possibly exist?

 

[Cheman]

I think a more interesting question would be to ask why does multiplication work or division?

ie - when we do multiplication, long and short, the is a set algorithm whihc we fulfil. eg - to multiply 234 by 5 we would times 4 by 5, etc. Same with long and short division.

I would be very interested to know why these algorithms actually work and whether they can be prooved.

Thanks in advance.

 

[matt grime]

Your asking to know why long mulitiplication works? Honestly? Or what multiplication means, ie why does it distribute. Well, for a useful introduction can i suggest

www.dpmms.acm.ac.uk/~wtg10[/URL]

there is a link to Gowers's informal writings, one explains why multipliation is commutative I think, though it is to do with sets.

 

[FulhamFan3]

ugh. These post are stupid. Even stupider is that people take these seriously and give long drawn out explanation.

He wants proof that two quantities put together make a bigger quantity? That's called an axiom. There is no proof because is a basic obvious thing. If someone does come up with a proof I'm going to ask for a proof of the basic statements they used to prove it. It's all got to start somewhere. At some point things are just obvious.

 

[Cheman]

"Your asking to know why long mulitiplication works? Honestly?"

Well, yes - a proof for long multiplication and division must clearly exist, as the methods we use are simply algorithms. eg - it is obvious that 2 + 1 =3; its an axiom, but when we want to work out what 456*34 is or 6879/43 there is a specific set method we follow. This is an algorithm. I was just wondering why these algorthims work?

Thanks.

 

[Curious3141]

"Your asking to know why long mulitiplication works? Honestly?"

Well, yes - a proof for long multiplication and division must clearly exist, as the methods we use are simply algorithms. eg - it is obvious that 2 + 1 =3; its an axiom, but when we want to work out what 456*34 is or 6879/43 there is a specific set method we follow. This is an algorithm. I was just wondering why these algorthims work?

Thanks.

Long multiplication ? Depends on distributivity of multiplication over added expressions :

The following is written out to correspond to the right to left (least significant to most significant) method taught in elementary school. When multiplying by a product of ten, shift one place left, by a product of hundred, two places left. Those should be obvious.

456*34 = 456*4 + 456*30

= (4*100 + 5*10 + 6)*4 + (4*100 + 5*10 + 6)*3*10

then just add it all up. The addition of remainders to the next most significant digit depends on associativity of addition.

Long division similarly depends on distribution i.e. \frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}. I'm sure you can see why.

 

[K.J.Healey]

So what you're saying is you can't prove addition?
Just kidding.

 

[JasonRox]

So what you're saying is you can't prove addition?
Just kidding.

Crabs is a very insightful person, who came up with such a beautiful question. :zzz:
Just kidding. I hope he/she stops smoking crack.

 

[MattTuc13]

The proof for addition is simple...

THE DEFINITION!

We define addition by adding numbers together to find their sum, therfore, that is the proof.

Descarte-Philosopher whose most known statement is "I think therefore I am".
Before he came to this conclusion, he said that the only thing that can be proven in the world is math. Why? Because by definition it is true.

What you are doing is trying to prove something that by definition is already true.

 

[HallsofIvy]

ugh. These post are stupid. Even stupider is that people take these seriously and give long drawn out explanation.

He wants proof that two quantities put together make a bigger quantity? That's called an axiom. There is no proof because is a basic obvious thing. If someone does come up with a proof I'm going to ask for a proof of the basic statements they used to prove it. It's all got to start somewhere. At some point things are just obvious.

An axiom? It's not even true. -7 "put together" with -4 (addition) is -11. That's not a "bigger quantity". all you are saying is that you do not understand the original question.

 

[FulhamFan3]

An axiom? It's not even true. -7 "put together" with -4 (addition) is -11. That's not a "bigger quantity". all you are saying is that you do not understand the original question.

math with negatives is equivalent to subtraction. the question is can you prove addition, not subtraction.

 

[metrictensor]

this is stupid

In your proof you have assumed addtion by applying subtraction. The problem is here that you don't understand what is meant by proof. My explanation will appear below.

1+1 = 2
subtract one from both sides
1 = 1

lol.

 

[Hurkyl]

math with negatives is equivalent to subtraction. the question is can you prove addition, not subtraction.

Actually, additive inverses are usually defined first, then subtraction defined in terms of that.

 

[metrictensor]

First off good question dr. Don't fret if 99% of the answers you get here are obviously from people who have never thought before. Short answer to your question is 'by definition'.

The problem, as I see it, is with what is meant by proof. All those mathematical proofs are useful within mathematics but what about in the everyday world. Ask a child what 1 + 1 is and they will say 2. How did they learn that? Your question is the same as asking 'why are all unmarried men bachelors?' The only reason is by definition. If you can understand 1 you can understand any number. Ask someone how many parents they have and they will say two. The reason 1 + 1 = 2 is the same as why the letter 'b' follows 'a'.

It is not that the mathematical explanation given above is "wrong". It is only appropriate in a particular context. How do we know this? In everyday life even people who have never seen the (so called) proof understand that 1 + 1 = 2. They know this by definition.

In most cases, anyway, the problem arises from a misunderstanding of the word proof. What is even more astonsihing is how little is gained when we answer this question. I understand perfectly the answer to your question. The benefit comes not from finding an answer but in the understanding we get from seeing how we went about finding the answer.