Proving Basics of Numbers: 2 + 2 = 4

[anantchowdhary]

Numbers are used to tell us about the quantity of something

Using just numbers and logic how do we prove that say 2+2=4

we can't use objects here as then the following problem comes up:
if we have say 3/2+1

and we take all numbers to be objects..the 3 objects divided by 2 objects is 3/2 +1 object which can't be resolved ...so please clear my doubt


this is very important as numbers form the basis of math!

Thanks

 

[HallsofIvy]

Using Peano's axioms for the natural numbers, (see http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/NUMBERS.pdf ) every number a has a "successor" s(a) and every number except 1 is the successor of some number. 2 is defined as s(1), 3 is defined as s(2), and 4 is defined as s(3).

Further addition is defined by
a+ 1= s(a). If b is not 1, then it is the successor of some number, say c: b= s(c). In that case a+ b is defined as s(a+ c). While that is a "recursive" definition, it can be shown that it defines unique "a+ b" for every a and b.

Now, 2+ 2= s(2+ 1)= s(s(2))= s(3)= 4.

If I remember correctly Russel and Whitehead, using concepts simpler than the Peano axioms, required 4 pages to prove 2+ 2= 4!

 

[anantchowdhary]

i have one thing more to ask

in physics...we just make use of numbers isn't it...i mean we can do all the operations and mathematical manipulations just because we can use these theorems of numbers isn't it...?

thanks for all the help

 

[anantchowdhary]

Isnt there any simpler way to prove these theorems...i don't really know the set theory and relations..etc

 

[HallsofIvy]

No, there is no "royal road" to mathematics! If you are interested in learning the proofs of the fundamental concepts of mathematics, then you will need to learn set theory, etc.

 

[anantchowdhary]

hehe...but well...people like pythagoras and co. wouldn't have been using these axioms to prove their theorems would they...!

Is there any primitive proof...which isn't so rigorous but will do for basics in calculus!

thanks

 

[CRGreathouse]

Pythagoras wouldn't have felt the need to prove 2 + 2 = 4. Modern mathematicians did, so we/they invented axiomatic set theory and a plethora of formal logics.

You can do as the ancients and take 2 + 2 = 4 on faith, or you can learn the basics of set theory (not hard!) to understand these proofs.

 

[anantchowdhary]

Could you please provide a link..as to where i would get the necessary theory needed to understand Peano's axioms

and about pythagora's :) don't we DEFINE addition and wouldn't he have thought of multiplication of numbers as different as compared to that of objects...like
3/2+1

and we take all numbers to be objects..the 3 objects divided by 2 objects is 3/2 +1 object which can't be resolved further?

thanks

 

[anantchowdhary]

one more thing...dont we define operations like say $$\frac{\frac{3}{2} +\frac{1}{3}}{2}$$ as being done imagining objects and getting the ans on each operation and then considering only the NUMBER part of the answer and carrying fwd the calculations

 

[gmax137]

anant - try to find David Berlinski's (sp??) book "advent of the algorithm." I am reading this now and it goes into some of these questions. And it is written (very) informally, not like a normal mathematical book. In fact, it may be a little too informal in spots.

 

[anantchowdhary]

anant - try to find David Berlinski's (sp??) book "advent of the algorithm." I am reading this now and it goes into some of these questions. And it is written (very) informally, not like a normal mathematical book. In fact, it may be a little too informal in spots.

Ill surely try to find the book..

id just lik to ask..is there anything wrong in my defiinition.As this explains the gr8 use of math we can do in physics...as we just take any equation(lik f=ma) and manipulate it according to our needs as anyway operating on two equal numbers will give us equal numbers,no matter what ,as the rules are same for bothe numbers,according to our definition.

thanks

 

[CRGreathouse]

id just lik to ask..is there anything wrong in my defiinition.

If you mean

"dont we define operations like say $$\frac{\frac{3}{2} +\frac{1}{3}}{2}$$ as being done imagining objects and getting the ans on each operation and then considering only the NUMBER part of the answer and carrying fwd the calculations"

then I can't make sense of it, let alone say that it's correct. If not, which definition?

 

[granpa]

the successor function obviously forms the basis of math but we could DEFINE the successor of 9 to be zero in which case we would have modular arithmetic. also we could define more than one successor for a given point. then we no longer have numbers but a graph of interconnected points. what field of math would that be?

 

[anantchowdhary]

If you mean

"dont we define operations like say $$\frac{\frac{3}{2} +\frac{1}{3}}{2}$$ as being done imagining objects and getting the ans on each operation and then considering only the NUMBER part of the answer and carrying fwd the calculations"

then I can't make sense of it, let alone say that it's correct. If not, which definition?

by this i mean...3/2 +1/3 using objects that is say using using an orange...=11/6

Now,if we take the case of $$\frac{ \frac {1}{12}}{3}$$
one onragen by twelve oranges is 1/12.Now i think of 1/12 as 1/12 oranges again and repeat the algorithm to get my answer...this is OK as a definition of an algorithm isn't it?

the successor function obviously forms the basis of math but we could DEFINE the successor of 9 to be zero in which case we would have modular arithmetic. also we could define more than one successor for a given point. then we no longer have numbers but a graph of interconnected points. what field of math would that be?

Yes obviously we could do this...but wouldn't make much sense

 

[granpa]

it wouldn't make sense for numbers. it would for a graph. I was just wondering what that field is colled.

 

[Diffy]

by this i mean...3/2 +1/3 using objects that is say using usnig an orange...i decide that the sum is 11/6..
and then i take 11/6 oranges and divide them by 2 so..i get 11/12 oranges...now is this algorithm correct?

You are not clear enough. I have no idea what you are asking.

Yes obviously we could do this...but wouldn't make much sense

But it does make sense! Modular arithmetic is at the heart of modern algebra.

 

[CRGreathouse]

by this i mean...3/2 +1/3 using objects that is say using usnig an orange...i decide that the sum is 11/6..
and then i take 11/6 oranges and divide them by 2 so..i get 11/12 oranges...now is this algorithm correct?

I have no idea what you're trying to say. It doesn't look like a definition to me.

Yes obviously we could do this...but wouldn't make much sense

I don't suppose you think that 1:00 comeing thirteen hours after 12:00 makes much sense, then?

 

[granpa]

since the successor function is the basis of math then maybe we shouldn't think in terms of one, two, and three but rather first, second, and third. 2+3 becomes the second after the third. 2*3 becomes the second third. its just semantics but it might make the underlying fundamental idea clearer.

what does three mean anyway? thirdness?

 

[granpa]

one more thing...dont we define operations like say $$\frac{\frac{3}{2} +\frac{1}{3}}{2}$$ as being done imagining objects and getting the ans on each operation and then considering only the NUMBER part of the answer and carrying fwd the calculations

division is defined as the inverse of multiplication. multiplication is defined in terms of (multiple) addition(s). addition is defined in terms of (multiple) successor function(s).

 

[anantchowdhary]

I have no idea what you're trying to say. It doesn't look like a definition to me.



I don't suppose you think that 1:00 comeing thirteen hours after 12:00 makes much sense, then?

Well this is just a SYSTEM used to measure time..isnt it...?i said it won't make sense to everything like in the case of day to day life objects...but of course we could define anything we like..
and what i was trying to say is ..that is it that we consider ever number as an object or a number of objects...in our present system of numbers...how else do you prove say $$\frac{1}{a} -\frac{1}{b}= \frac{b-a}{ab}$$

Thanks

 

[anantchowdhary]

Please comment!
Isnt the basic math we use..defined on day to day use...
thus 2+2=4!

and so on...

 

[granpa]

successor function=s(n)=n+1

1=1
2=1+1
3=1+1+1
4=1+1+1+1

2+2=(1+1)+(1+1)=1+1+1+1=4

 

[HallsofIvy]

Well this is just a SYSTEM used to measure time..isnt it...?i said it won't make sense to everything like in the case of day to day life objects...but of course we could define anything we like..
and what i was trying to say is ..that is it that we consider ever number as an object or a number of objects...in our present system of numbers...how else do you prove say $$\frac{1}{a} -\frac{1}{b}= \frac{b-a}{ab}$$

Thanks

Since you have never said what you mean by an "object" I don't know how to answer that.

If you simply mean that "1+ 1= 2" is defined by "one actual physical object plus another actual physical object equals two physical objects", no, that is not how we define "1+ 1= 2". That is an application of mathematics not mathematics itself.

Please comment!
Isnt the basic math we use..defined on day to day use...
thus 2+2=4!

and so on...

No, it isn't. You are not comprehending the distinction between "mathematics" and "an application of mathematics".

And, indeed, if you thought so,why in the world would you say, in your first post, "Using just numbers and logic how do we prove that say 2+2=4?"

 

[anantchowdhary]

Since you have never said what you mean by an "object" I don't know how to answer that.

If you simply mean that "1+ 1= 2" is defined by "one actual physical object plus another actual physical object equals two physical objects", no, that is not how we define "1+ 1= 2". That is an application of mathematics not mathematics itself.


No, it isn't. You are not comprehending the distinction between "mathematics" and "an application of mathematics".

And, indeed, if you thought so,why in the world would you say, in your first post, "Using just numbers and logic how do we prove that say 2+2=4?"

Umm..see when numbers were invented i don't think anyone would have thought that 2+2=4 on the lines of axioms or basic definitions...maths came from physical objects isn't it?

Where could i find the definitions of operations like addition ,and other basic operations from?

and regarding the thing that 'why in the world' would I have said that..is that what i think is(and believe)is that numbers came and are certainly to do with physical objects arent they..?

and also..how would you say that $$\frac{1}{a} -\frac{1}{b}= \frac{b-a}{ab}$$


thanks for all the help...

 

[HallsofIvy]

Umm..see when numbers were invented i don't think anyone would have thought that 2+2=4 on the lines of axioms or basic definitions...maths came from physical objects isn't it?

Yes, and human beings came from monkeys. That doesn't make them the same thing. Are you talking about mathematics or are you talking about counting?

Where could i find the definitions of operations like addition ,and other basic operations from?

I thought I had answered that. The difficulty seems to be that you want definitions to fit YOUR idea of mathematics- and you won't find that anywhere.

and regarding the thing that 'why in the world' would I have said that..is that what i think is(and believe)is that numbers came and are certainly to do with physical objects arent they..?

I can't speak for what you think and believe but that is certainly not what mathematicians today think and believe.

and also..how would you say that $$\frac{1}{a} -\frac{1}{b}= \frac{b-a}{ab}$$


thanks for all the help...

This is very strange. You initially protested that the "definition" of numbers in terms of single "objects" would not lead to fractions. I, basically responded with a more abstract definition that would lead to fractions. Now you are insisting that numbers MUST be defined in terms of "objects" and still complaining that you can't get fractions out of them! Yes, as long as you insist that numbers be defined in terms of counting "objects", you cannot get more than integer arithmetic out of them. That is one reason numbers are NOT defined that way. How they might have been thought of historically is not relevant.

 

[gmax137]

I don't agree with Anant that numbers "come from" objects. I have two eyes, but I can use them to look at you twice - so are "looks" objects? What if I "think twice?"

On to fractions: Let's say I have an apple. OK, "one" apple. If I cut in in half to share with you, what have I done? I have "two" "halves." Most natural...

 

[anantchowdhary]

I don't agree with Anant that numbers "come from" objects. I have two eyes, but I can use them to look at you twice - so are "looks" objects? What if I "think twice?"

On to fractions: Let's say I have an apple. OK, "one" apple. If I cut in in half to share with you, what have I done? I have "two" "halves." Most natural...

I'm sorry i don't agree.Numbers originated with the need to describe a quantity.So the the algebra we do..is purely on definition on operations isn't it...and those operations on numbers are similar to that on physical objects...like $$\frac{3}{4}+\frac{1}{4}=1$$, just like that for any physical object
This is what i meant..

thanks

 

[Alfi]

I thought this was a good book for me to re-learn

Mathematics: From the birth of numbers / Jan Gullburg
ISBM 0-393-04002-x

I thought he did a nice quick history.

 

[HallsofIvy]

I'm sorry i don't agree.Numbers originated with the need to describe a quantity.So the the algebra we do..is purely on definition on operations isn't it...and those operations on numbers are similar to that on physical objects...like $$\frac{3}{4}+\frac{1}{4}=1$$, just like that for any physical object
This is what i meant..

thanks

?? But in your very first post

Numbers are used to tell us about the quantity of something

Using just numbers and logic how do we prove that say 2+2=4

we can't use objects here as then the following problem comes up:
if we have say 3/2+1

and we take all numbers to be objects..the 3 objects divided by 2 objects is 3/2 +1 object which can't be resolved ...so please clear my doubt


this is very important as numbers form the basis of math!

Thanks

You said "we can't use objects here". Yes, if you insist that numbers always refer to the quantity of objects, then you can't solve problems like that. That is why number do NOT always refer to the quantity of objects and why a basic definition of "numbers" cannot involve "objects". Another reason is that we use number for measuring lenght, area, volume, etc. and our definition of "number" has to be abstract enough to apply to all of those.

 

[anantchowdhary]

Yes I agree...that numbers should be abstract...but the thing is at school level we just arent taught any of the 'definitions' for any mathematical operations...and the calculations can be carried out usnig thought experiments inolving objects...
so in the first post i was just referring to a proof without objects...which u supplied but i cudnt understand as iv not much knowledge of the set theory..so i seem to have reverted to objects :)