It's obvious that 2 + 2 = 4, but how can you prove this?
Is it possible to prove it?

It's obvious that 2 + 2 = 4, but how can you prove this?
Is it possible to prove it?
You need to set up an axiom system which includes the definition of addition, the definition of numbers, etc. ZermeloFrankel is the usual system.Originally Posted by IAlexN
Yes. It follows from the construction of the natural numbers and the operation of addition.Originally Posted by IAlexN
You can see a proof in Landau's Foundations of Analysis.
You can also see this done in Rudin's Principles of Mathematical Analysis and other textbooks as well.
While matheman's statement is correct, the Zermelo Fraenkel axioms (plus choice) are usually presented in formal treatments of mathematical logic, while in the ordinary course of things arithmetic is developed from the somewhat less formal Peano axioms. The Peano axioms are actually included in the Zermelo Fraenkel axioms.
BTW the construction of the arithemetic of the integers is interesting from a technical perspective, but also incredibly tedious and boring.
2 = 1 + 1
4= 1+1+1+1
thus 2 + 2 = 1+1+1+1 = 4
Simples
Actually the real proof is more along the lines ofOriginally Posted by smokey
4=(3+1)=((2+1)+1)=(2+(1+1)=2+2
But first one very carefully defines what n+1 means for any n.
Natural numbers have a specific meanings, in which they follow the peano axioms. if it does not follow it, then it will no longer be a natural number e.g. 2,4..
so it goes like this;
firstly it starts off with "", then........... so on and on.
so really deduces pretty much in to;
as
thereby completing the proof.
this is actually something i learned last year when i asked my teacher, how do you prove 1+1=2, and he started talking about Peano Axioms, and gave us more of the proof including larger natural numbers. im not sure if remembered the steps correctly, but hope it helps.
If you look closely, that is pretty much what I just said, without getting into the technical notion of a "successor".Originally Posted by Heinsbergrelatz
well then in that case, i repeated the proof with the successor.If you look closely, that is pretty much what I just said, without getting into the technical notion of a "successor".
Well, in my opinion the Peano Axioms are rather circular, and therefore the conclusions we gain from them shouldn't be regarded as absolute. I believe that the arithmetic of all numbers is arbitrary, merely a law of thought made by god that can't be proven rigorously without basing it on some other axioms which themselves can't be proven, unless based on some other axioms which can't be proven due to the same reasons as the others, and so on.
Can you point out where they refer to themselves?
No. You don't understand my post. The point is the following: if you can't prove the Peano axioms, than it's similarily incorrect to say that you can prove other statements based on them.Originally Posted by MagiMaster
You are quite correct, but you are missing the point. Axioms are the starting point, not something to be proved.Originally Posted by Ellatha
I don't believe I'm missing the point; I am aware that axioms are merely provide a base from extentions of number theory. It is only when people claim to have "proved" these simple arithmetic predicates that I believe this misconception needs to be addressed. Of course, you and I know that one plus one equals two, it is just the way things are (unless god changes thought's laws, I believe), it's only that this can't be proved until one can axiomatically define n + 1.Originally Posted by mathman
What you are describing is not circularity.Originally Posted by Ellatha
It is the very nature of axioms.
Axioms, any axioms, are by definition, statements that one accepts as true without proof. In essence they are accepted because they are "obvious".
http://en.wikipedia.org/wiki/Peano_axioms
The Peano Axioms are in essence nothing more than the assumption that the natural numbers exist and are what one normally thinks they are plus the axiom of induction. They are usually used in an introductory treatment of the foundations of mathematics.
From the Peano Axioms one can construct all of the rest of the number systems usually encountered in mathenatics  the integers, the rational, the reals and the complex numbers. No circularity.
If one want to be very formal, one uses the ZermeloFraenkel axioms. The Peano axioms are actually contained in the ZermeloFraenkel axioms. The real difference is that the ZF axioms are formulated within the very strict structure of axiomatic logic and set theory. To ZF one usually adds the axiom of choice, and ZF plus choice is the foundation of essentially all of mathematics. A notable exception is what is called the school of constructive mathematics, established by Errett Bishop, and in that school one does not assume the axiom of choice.
"God made the integers, all else is the work of man." – Leopold Kronecker
One thing depends on another here and the dependant could be wrong making the system itself incorrect regardlessthis is circular.Originally Posted by DrRocket
Axioms are independent of proofsthey depend on nothing actually, which makes them more like the complete opposite of what I'm describing.Originally Posted by DrRocket
It's not just because they're obviousthey can't be proven. Reality is not intuitive, it is rigorous.Originally Posted by DrRocket
One needs to assume that the Peano Axioms are correct here, and assuming is not the nature of mathematics, it needs to be proven. It is only in our reality that the Peano axioms hold true because 1 + 1 = 2 and so on, but at the same time this can be realized without any set of axioms.Originally Posted by DrRocket
"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality."Originally Posted by DrRocket
Albert Einstein
At some point, you've got to stop asking "why?", and for mathematician's, who want to be rigorous, they codify their stopping point by saying "These axioms are our stopping point, and we'll make as few of them as possible." Of course, you're free to define your own axioms and see what you can make from them, but unless you come up with something very interesting, or some smaller set of axioms that can be used to prove the existing axioms, no one really cares.
I don't really see how your arbitrary "God did it" explanation is any better. (Saying 1+1=2 is obvious is basically making it an axiom.)
BTW, none of the Peano axioms can be proven using the remaining axioms, so I can't see how anything here is circular either.
No.Originally Posted by MagiMaster
Everything has a reason. Your "we don't know the answer, so we will just accept it" is by far more arbitrary than saying "god did it."Originally Posted by MagiMaster
I don't need to define any axioms, nor do I want to. These socalled axioms, or things that can't be proven, to me, are the bases of mathematics from which everything is derived, I believe god made these bases, and everything else ramified from that.Originally Posted by MagiMaster
It provides a viable explanationit's obvious that in the Universe there is a tremendous amont of order, and anybody that has studied probability theory knows that there's no way that it got there by accident, I simply believe that these initial conditions that nobody can really prove needed to be put there by some higher order. It is far better than your explanation because it doesn't run from the problem.I don't really see how your arbitrary "God did it" explanation is any better.
There is a reason for thatyou don't understand my explanation.so I can't see how anything here is circular either.
Yes, you can prove it experimentally. You need apples and a table. Put two apples on the table. Then add another two.Originally Posted by IAlexN
No Ellatha it is not spam.Originally Posted by Ellatha
Your confidence is outpacing your competence and understanding again.
Better rein it in and learn something.
No the reason that magimaster does not understand your explanation is that your explanation is seriously flawed.Originally Posted by Ellatha
You need to learn to keep religion separate from science and mathematics.
They are quite compatible. But they are also disjoint.
It has very little to do with religion in the conventional senseI'm not referring to Catholicism or anything of that sort, but the structure of the universe. One can make reference to god without actually bringing up such matters.Originally Posted by DrRocket
How would you know how competent I am? You know very little about meyou also are surprisingly dismissive for an analyst, given this type of mathematician specializes in theory yet you take seemingly no time to explain anything.Originally Posted by DrRocket
I can tell quite a bit from your posts and your PMs, based on a lot more experience with people and their understanding of mathematics than you have at this point in your life. I have seen a lot of smart ass little kids.Originally Posted by Ellatha
Your inability to understand an explanation is not proof that there is no explanation. Sometimes I deliberately leave a few steps for you to figure out  it would do you good to take advantage.
When I learned real mathematics, the professor did not explain things to us. He expected us to figure it out and present the results to the class  no references or consulting with anyone allowed. When you have to figure it out for yourself you tend to understand what you learn. The down side is that sometimes that method is a bit slow, so you don't get quite as much breadth at first. You pick it up later.
You learn mathematics by learning how to think about mathematics and by doing mathematics. You do not learn it by having someone teach you how to solve certain problems by prescribed methods. There are no rules on how to solve mathematics problems, only constraints in the form of adherence to logic. There are a lot of right ways to address a problem, and learning mathematics is learning how to discover one or more of them.
ALL mathematicians specialize in theory. Analysts specialize in analysis  which basically is "on beyond calculus". Other mathematicians specialize in other types of theory  algebra, topology, geometry, combinatorics, etc. But really most mathematicians use bits and pieces from all branches of mathematics. Functional anlaysis, for instance uses algebra, topology and other pieces of analysis simultaneously (see "topological vector spaces"). Any decent mathematician can teach any undergraduate class and most firstyear graduate classes.
Before you start lecturing PhD mathematicians on the subject of mathematics, I suggest that you first, at the very least, understand calculus well enough to be able to write a calculus book off the top of your head  with proofs.
You have some serious misconceptions regarding mathematics and they need to be corrected for the benefit of other readers.
You can learn from the corections or not, as you choose. If you want to be a smart ass, then that is OK, but in that case I am done with you.
Originally Posted by Ellatha
Such references are frought with peril in discussions of physics, and completely irrelevant to mathematics.
Well, according to you I am undercompetent, bad at math, egotistical, no close to nothing about calculus. I'd hardly say that I'm the one being egotistical.Originally Posted by DrRocket
Where was my "inability to understand an explanation?"Originally Posted by DrRocket
No. All you do is say a one or twoword reply that does absolutely no goodwhatsoever in the form of "nope," "wrong," or the like.Originally Posted by DrRocket
I'm sure your professor's lectures weren't composed of a "nope" than an exit to the door.Originally Posted by DrRocket
Sure, but clearing up misconceptions with actual explanations is undeniably far better than dismissal.Originally Posted by DrRocket
I know what an analyst is.Originally Posted by DrRocket
Well, perhaps if you understood that this doesn't have as much to do with math as it does with reality than you wouldn't be so quick to jump to your credentials and use your logic.Originally Posted by DrRocket
"Serious misconceptions?" Much of what you say is incorrect I have already confirmed by professors at reknown universities (including M.I.T.), for the most part, I find that you tend to overexaggerate small statements that people could potentially confuse and other such detail into blown out complete misunderstanding of a concept.Originally Posted by DrRocket
Sure thanI don't have a problem finding a guide that will actually help me to reach my full potential rather than somebody that tells me how dumb I am all of the time.Originally Posted by DrRocket
Besides, I generally find that people with advanced degrees are not nearly as intelligent as they believe themselves to be and, for the most part, become masters of trivia rather than accomplishing anything noteworthy.
And only goes to show that your consistent applying it to math is incorrect.Originally Posted by DrRocket
One can not use a counting argument as a mathematical proof since there is a possibility that the individual miscounted.Originally Posted by Twit of wit
[quote="Ellatha"]Wrong.Originally Posted by DrRocket
Though sometimes when they were drinking coffee they just shook their head from side to side.
You are rather sure of a lot of things that just are not so.
@Ellatha, you say you don't have to stop asking why, yet you say "god did it" and then stop asking why (which is exactly as arbitrary as declaring such an axiom but without the religious overtones). And anything you accept on the basis of something other than a proof is an axiom to you. You said 1+1=2 is just part of how the universe is, which means that you accept 1+1=2 as an axiom.
(In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either selfevident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.)
I never said I stopped asking why, nor do I claim to absolutely know the complete truth of the scheme of things. This is merely something you put in there to support a poor rebuttale based on some sort of pseudohypocrisy. Unlike you and others that support the idea of axioms, I believe they came there for a reason, and I provide this reason blantaly. This is how are opinions differ. If you need further evidence of this, it can be found in this simple question: why are there such axioms? You can explain to me why the square root of the sum of the squares of the legs of a right triangle equal the hypotenuse, and that the circumference of a circle is the product of pi and twice the radius, but not the existence of the axiom.Originally Posted by MagiMaster
Rather than actually trying to lclaim I don't know what is going on here or am changing my view point repetitively, it would do you best to address why you choose to support your view point.
It is NOT true that "the the square root of the sum of the squares of the legs of a right triangle equal the hypotenuse, and that the circumference of a circle is the product of pi and twice the radius". That is only true if one accepts the axioms of Euclidean geometry. Those axioms are assumptions, pure and simple. There are other geometries in which these thiings are not true.Originally Posted by Ellatha
Moreover, since you have some oddball notion of "reality", the real geometry of the surface of the Earth and of the univese itself is NOT EUCLIDEAN.
And you quite clearly don't know what is going on here. The real problem is that you don't know that you don't know what is going on here.
Mathematics does not answer the question of WHY anything is true. It only evaluates the logical implications of a set of axioms which are assumed "true" for the purpose of further argument. Mathematics need not address "reality". Sometimes it does, through the intervention of physics and sometimes it does not. Reality is the provence of science, not mathematics.
In fact science does not explain WHY nature works the way that it does but only attempts to explain HOW nature works, in the form of predictions based on some known set of conditions.
HOW is the provence of science. WHY is the province of philosoophy and theology. It is best to keep them separate.
Why are there axioms? Because we have to assume something is true. If you assume that nothing is true, you really have no ground to stand on to work out anything else.Originally Posted by Ellatha
Saying "god did it" just assumes it's true and attaches some extra baggage on top of that. It's not a proof. It doesn't derive from any simpler truths. It doesn't really do anything substantially different from stating something as an axiom. It answers why for you, but not for anyone who doesn't share your beliefs. On the other hand, saying something is an axiom doesn't attach the baggage of why. It simply states that the person working has chosen this as a starting point.
The Peano axioms, for example, are accepted as a common starting point for a number of reasons.
Originally Posted by MagiMaster
1. They work
2. They formalize the notion of natural numbers which are acceptable to animals who can count their fingers, whcih seems to include all mathematicians, most baboons, a significant percentage of college freshman and the occasional high school student.
Everybody bar the Indiana legislature I would guess2. They formalize the notion of natural numbers which are acceptable to animals who can count their fingers, whcih seems to include all mathematicians, most baboons, a significant percentage of college freshman and the occasional high school student.
This is not an appropriate answer. You're answering the question "how do the concepts of axioms, unproven theorems, aid mathematicians to build upon them?" rather than "why are there axioms in the universe?" The answer to my question should not be artificial in the sense that it relates to what we have done to make things easier for our selves, eg., notations etc., but why in terms of how proofs aren't enough for these unprovable theorems.Originally Posted by MagiMaster
It provides us with a very important answer: a reason, something you and those like you simply can't provide. Apparently, unlike you, I'm one of the few members on this board that believes that everything in the universe has a reason in the metaphysical sense.Originally Posted by MagiMaster
I agree, but I also believe that there are answers out there, and for those who don't want to continue to search for them than I couldn't care less whether they threw in the towel or not. Mathematics is not a perfect system, and I believe that since Kurt Godel's Incompleteness Theorem it has been demonstrated that finding the answer to every question is not possible with mathematics alone (we need philosophy, metaphysics, and other such studies that have long been abandoned due to their lack of applications). It seems that the more a certain discipline tends towards gathering implications about the universe, the less its applications. For example, science provides us with more applications than mathematics or philosophy, but with less answers, while philosophy tends to provide us with more answers than mathematics or science, but is far less applicable because of its focus on discourse. Of course, the two previous statements are highly debatable and I suggest they be taken with a grain of salt, but at least the ideology of the paragraph as a whole can be understood in terms of my view point here.Originally Posted by MagiMaster
Could someone explain what a successor basically is in (or means) in mathematics, as I haven't read about it before, and am very curious.Originally Posted by DrRocket
Now you're getting to the point. What you're talking about isn't mathematics, therefore you shouldn't be calling it mathematics.Originally Posted by Ellatha
@IAlexN, a successor is kind of arbitrary, but if you start from the ideas that:
 0 is a number
 Every number has a successor (a number that comes after it)
That's all you need to define the (nonnegative) integers and from there you can define addition, multiplication and a few other things. A couple more definitions and you can get the reals and just about everything else. (Those are numbers 5 and 6 out of the 9 Peano axioms.)
and to add to this, you get the negative integers by showing that there is a predecessor for every number.Originally Posted by MagiMaster
I recall an attempted proof by Ludwig Wittgenstein. It was something along the lines of:
a^bc = (a^b)^c
(a^2)^2 = a^2*2 = (a^2)^1+1 = a^2 * a^2 = a^1+1 * a^1+1 = (a*a) * (a*a) = a*a*a*a = a*4
"attempted" is correct.Originally Posted by AlphaMuDelta
Showing that 2 + 2 = 4 is not so much a proof as it is a definition of what means by the operation "+' and by the number 4.
Without becoming pedantic, 4 is actually defined as 3+1. So, again, the proof goes something like 4=3+1=(2+1)+1=2+(1+1)=2+2.
To really understand this you need to go through the construction of arithmetic of the integers based on the Peano Axioms. See for instance Foundations of Analysis by Landau.
lol i wonder if this is what we are doing in our heads as young children when we are 2 or 3 and learning how to add simple single digit equation's
Kind of intriguing to wonder if our minds as babes are that complex (kind of straying into psychology sry)
At that point, it's mostly rote memorization and visualizing piles of objects.
True.Originally Posted by DrRocket
I can not use a mathematic proof since there is a possibility that the individual is an idiot.Originally Posted by Ellatha
Yes. The point of a proof is to prove that something is true with 100% certainity. Counting cannot do this.Originally Posted by Twit of wit
If you are still not convined, I offer you the following challenege: explain to me (no need for a rigorous mathematical proof) why one plus one equals two, without using circular reasoning but using valid logic.
Well, I think Ellatha is actually right. And the others are wrong.
You can not proove that 2 + 2 = 4 is true.
You can only proove that 2 + 2 = 4, if the axioms from ZermeloFrankel are true.
"Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see Axiomatic set theory and NonEuclidean geometry for examples)."Originally Posted by MagiMaster
from wikipedia
You don't have to assume anything. You don't have to assume something is true, neither do you have to assume something is untrue. And still you can work out proofs out based from a set of axioms.
And sometimes you can proove that those axioms are wrong.
Like Euclidean geometry, we all thought for thousand of years that those axioms were true, and now we have proved that one is wrong.
I know that I think
I think that I don't know much more.
If you have a set of axioms that you can prove are wrong, then using those axioms you can prove ANYTHING, including that the axioms are true.Originally Posted by kasper90
The reason is this. All proofs based on a set of axioms are reducible to logical sentences of the form "If Axioms A,B C, ... are true then X is true".
Now, a fundamental property of logic is that any false premise implies any conclusion whatever. This is simply how truth tables work. The sentence If A then B is always true if A is false.
So, if you have an axiom that you know is false, the statement "If A then X" is true no matter what X is. So in fact you say "If A then all of the axioms are true" and since all proofs witin the axiom system are of this form you have proved that the axioms are true.
This is an example of something that goes back to Godel's incompleteness theorems. He showed that it is impossible to prove, within the set of axioms using ordinary logic, that the ordinary axioms of arithmetic are consistent, unless they are inconsistent. So, paradoxically, if you can prove, within the axioms using ordinary logic, that the axioms are consistent, you have in fact proved that they are inconsistent.
BTW, you and Ellatha are both wrong.
Set theory sets out from certain conceptions such as "set", "element" and "property", which we are able to associate more or less definite ideas, and from certain simple axioms, which, in virtue of these ideas we are inclined to accept as "true". Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of the "truth" of the individual propositions is thus reduced to one of the "truth" of the axioms.
Now it has long been known that the last question is not only unanswerable by the methods of set theory, but that it is in itself entirely without meaning. We cannot not ask wether it is true that two equals sets have the same elements. We can only say that set theory deals with things called "equal sets", which is ascribed the property of containing the same elements. The concept "true" does not tally with the assertions of set theory, because the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; set theory, however is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.
It is not difficult to understand why, in spit of this, we feel constrained to call the propositions of set theory "true". Set theoretic ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of these ideas. Set theory ought to refrain from such a course, in order to give to its structure the largest possible logical unity.
If we now supplement the propositions of set theory by the single proposition that the measurement of visible objects on a table is represented by a member of the set of positive integers, the propositions of set theory then resolve themselves into propositions on the value of the measurement of the quantity of an object.
Set theory which has been supplemented in this way is then to be treated as a branch of physics (or a every day task to make sure that there is enoug cutlery on the table ). We can now legitimately ask as to the "truth" of set theoretic propositions interpreted in this way, since we are justified in asking wether these propositions are satisfied for the real things we have associated with the set theoretic ideas. In less exact terms we can express this by saying that by the "truth" of a set theoretic proposition in this sense we understand its validity with ruler and compasses.
Of course the conviction of the "truth" of set theoretic propositions in this sense is founded exclusively on rather incomplete experience.
With a little help from Albert Einstein
Originally Posted by kasper90
You are at least in part confusing mathematics with mathematical models as used in physics.
In mathematics (and this is the mathematics forum not the physics forum) axioms are statements that are assumed to be true without any further justification. Any application to "reality" or to the physical world is an assumption and application that lies outside of mathematics. There is no objective determination of the truth of axioms and no meaning to "truth" of the axioms outside of the axiomatic structure, but within that structure the axioms are "true".
The question of the truth of the axioms is not a mathematical question. Axioms are, by definition, taken to be true. What you cannot show is that any set of axioms that is sufficiently rich to support ordinary arithmetic is selfconsistent, unless it is in fact inconsistent. This is Godel's second incompleteness theorem.
It is quite easy to talk about the equality of sets, In fact all that you need to know is that "=" and "is" are synonyms.
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