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#1
04-18-2006, 08:28 AM
 sverresverre Guest Join Date: Apr 2006 Posts: 3
1+1 = 2 ?

My question is simple:

Can you prove that 1+1 = 2?

The answer everyone has ever given me, has been a long gaze, but the very fundaments of mathmatics is what makes me sceptical to science as a basis for everything. We have a strong scentiment that we can explain everything rationally (and if you are religion you tend to like to explain everything irrationally).

But since I can not see how one could prove 1+1 = 2, then science, like any other dogma is based on faith, and therefore I am automatically sceptical.

I would truly apprechiate some answers to my query, on the one place in the world where I think this question might be handled seriously and with as little mockery as possible (please).
#2
04-18-2006, 08:46 AM
 chrisk Charter Member Join Date: Nov 2003 Location: Southern ontario Posts: 6,568
Quote:
 Originally Posted by sverresverre My question is simple: Can you prove that 1+1 = 2? The answer everyone has ever given me, has been a long gaze, but the very fundaments of mathmatics is what makes me sceptical to science as a basis for everything. We have a strong scentiment that we can explain everything rationally (and if you are religion you tend to like to explain everything irrationally). But since I can not see how one could prove 1+1 = 2, then science, like any other dogma is based on faith, and therefore I am automatically sceptical. I would truly apprechiate some answers to my query, on the one place in the world where I think this question might be handled seriously and with as little mockery as possible (please).
Rough proof via peano arithmetic:

A few simple definitions in peano terms. 1 is the sucessor of 0, or S0. 2 is the successor of 1, or SS0

Any equation is equal to itself, so we can start with S0 + S0 = S0 + S0 (1 + 1 = 1 + 1)

Any expression of the form A + SB is equivalent to SA + B, so S0 + S0 is equivalent to SS0 + 0. Thusly we can prove that (1 + 1 = 2 + 0)

Since the expressions X + 0 and X are of equal value (by the Peano definition of addition,) SS0 + 0 can be reduced to SS0. Thus, we have proved that 1 + 1 = 2 according to Peano's axioms. (I think... hopefully I haven't skipped over anything or taken an illegal shortcut. It's been a while since I've worked with Peano.)

Algebraically, we can approach the problem somewhat simpler:

Take 1 + 1 = 2

subtract 1 from both sides, and you see that statement is equivalent to 1 + 1 - 1 = 2 - 1

Simplifying both sides, you can get 1 = 1, which is pretty self-evident.

Basically though... the statement is nearly so trivial as to be true by definiton. When we use the numeral 2, we are using it as a symbol to represent the number we get when we add 1 and 1 together... or else what the heck is it supposed to mean?

Does that help at all??
#3
04-18-2006, 08:53 AM
 Thudlow Boink Charter Member Join Date: May 2000 Location: Lincoln, IL Posts: 24,083
I hesitate to post this, because I suspect that we can come up with as good or better answers here on the Dope (like chrisk has), but here's one relevant link.
#4
04-18-2006, 08:54 AM
 nd_n8 Guest Join Date: Nov 2005 Location: Central Indiana, mostly. Posts: 1,593
:: gives sverresverre a long gaze ::

Scientific proof of 1 + 1 = 2, ok, try this simple experiment. Take one bottle cap, observe that it is in fact one bottle cap and place it on a blank piece of paper. Take another bottle cap, observe that it is in fact one single bottle cap and place it on the paper next to the first bottle cap. Double check your observations to ensure that you should have one bottle cap sitting next to another bottle cap. Double count the bottle caps on the paper to ensure that there are, in fact, two bottle caps on the paper. As a control, place a seperate blank sheet of paper to the side and do not place any bottle caps on it. Count how many bottle caps are on the second one and record this. Repeat this experiment several times, then compair the results.

Science is based on observable* conditions that can be repeated. If an object, fact, condition, etc. fails to meet those two conditions then it is not scientific fact. This is the dogma of science, it must be observable and must be repeatable, thus no faith is required.

*Note that many scientific facts, objects, conditions, etc. are not observable but their effects on their surroundings are. If the observations of the effects are repeatable and consistant then the fact, object, condition, etc. is theoretical. For example sub-atomic particles, black holes, and Brittany Spears' musical talent are all unobservable thus theoretical.
#5
04-18-2006, 08:57 AM
 Grey Guest Join Date: Sep 2002 Location: Ottawa, Canada Posts: 6,381
This is going to move to GD fairly quickly. Basically you're asking if there are un-provable statements within mathematics that make it inconsistent. If math is inconsistent then anything that uses mathematics to model empirical data must also be inconsistent.

The thing is there are “unprovable” statements in mathematics called axioms. It's from these axioms that the resulting structures hang. You can have a variety of mathematical structures, of course, all based on differing starting axioms. The thing is, within those structures the mathematical reasoning is consistent, even though there maybe statement not provable within that framework.

But in the end, if I give you 1 apple and then I give you another apple you've now got 2 apples.
#6
04-18-2006, 09:01 AM
 Alessan Guest Join Date: Jul 2000 Location: Tel Aviv Posts: 22,412
Math is a language. 1+1=2 by definition.
#7
04-18-2006, 09:03 AM
 Oy! Guest Join Date: Oct 2002 Location: Trenton, NJ Posts: 4,763
sverresverre, you are comparing apples and oranges.

1 + 1 = 2 because addition is defined such that 1 + 1 = 2. Mathematics is a human construct, that has no external reality whatsoever. It is a thing of symbols only - have you ever held a 1 or a 2 in your hand? No, because there is no such thing as a 1 or a 2. It's not a matter of taking it on faith that 1 + 1 = 2. It must be so because we have defined addition in that way. What math does is to postulate a certain set of "operations" such as addition, substraction, etc, and a set of rules.

Now, the interesting thing about this totally artificial construct mathematics is how useful it is in describing reality. We have never seen a situation that could be represented by 1 + 1 that did not equal 2. We don't expect we ever will. So mathematics becomes an excellent predictor of what will happen under many different circumstances. Probably the vast majority of technology you use was developed using mathematics. Why? Because this set of rules and operations that human beings came up with fits our experience so completely. But it is still an artificial construct, and we know this. So we know, based on the millenia of experience we have with it, that correctly applied mathematics is extremely likely to conform to external reality. We call this modelling, and although the term is recent, the concept is very old.

Religion is describing an external reality. There either is a god or there isn't. There either was a Jesus or there wasn't. And so on. It is not, nor does it purport to be, an abstract set of symbols that accurately reflect external reality. It is external reality, or at least that is what it claims to be.

External reality lends itself to evidence, to experiential proof. You know the table in front of you is solid (at the macro level) because you can rest your hand on it, and so can any other person who tries (assuming they have a hand). In essence, that what the scientific method is. It postulates something: the table is solid. And then it sets up replicable tests: you rest your hand on the table, and so does the next guy and the next and the next. Anyone who attempts to rest his/her hand on the table can do so. And other people can see them doing so. There is evidence.

Mathematics isn't the same kind of thing. It can't be seen, felt, heard, or sensed in any way. It's simply a humanly defined tool, a set of abstract concepts that, thus far, have accurately corresponded to our experience of external reality. There's no faith about it, except the faith that because it has always done so in the past, it always will correspond to external reality. If, in fact, it fails to do so, then the chances are that the abstract definitions we created and called mathematics are simply wrong, and need to be re-defined.

Religion claims to be external reality. It therefore lends itself to evidence, proof. Mathematics is a symbolic construct. They are not comparable.
#8
04-18-2006, 09:31 AM
 ultrafilter Guest Join Date: May 2001 Location: In another castle Posts: 18,988
Here's a sketch another proof that I think is a bit clearer:

You have a number 0, and you have an operator ' such that whenever n is a number, n' is too. Define + as n + 0 = n and n + m' = (n + m)'.

Now, what are 1 and 2? Let's call them 0' and 0'', respectively. 0' + 0' = (0' + 0)', and 0' + 0 = 0'. So 0' + 0' = 0''. In other words, 1 + 1 = 2.

I'm glossing over quite a bit here. For the full proof, I would have to define what 0, ', and = mean, but I don't think that's of any pedagogical value here.
#9
04-18-2006, 09:35 AM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 12,834
Quote:
 Can you prove that 1+1 = 2?
If we are talking in mathematics, then we need to agree on what a few things mean, including "prove", "1", "+", "=" and "2".

chrisk has given a good outline of a proof in which 1 is defined as S0 and 2 is defined as S1. However, that's not the only possible axiom system for the natural numbers, and in other axiom systems, the proof would be different. So Oy! has assumed a system in which "2" is defined as being "1+1", and of course in that system the proof that 1+1 = 2 gets to be a lot shorter.

At another extreme, Alfred North Whitehead and Bertrand Russell's book Principia Mathematica notoriously says on p. 360: "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." That is, PM takes 360 pages building up an axiomatic system for the natural numbers, at the end of which you can say that you have proven that 1+1 = 2. (I've only skimmed PM, but I can assure you that those 360 pages are heavy reading indeed.)
#10
04-18-2006, 11:05 AM
 Cervaise Guest Join Date: Mar 1999 Location: underpants Posts: 19,743
Quote:
 Originally Posted by sverresverre Can you prove that 1+1 = 2?
That's nothing. Can you prove that .999... + .999... = 2?
#11
04-18-2006, 11:10 AM
 Mathochist Member Join Date: Feb 2004 Location: New Orleans, LA Posts: 3,031
Quote:
 Originally Posted by sverresverre My question is simple: Can you prove that 1+1 = 2?
The current dominant philosophy, either explicitly or implicitly, is "structuralism". That is, the natural numbers are a structure (described by the Peano axioms mentioned elsewhere in the thread) which can be instantiated in various models (like the set theory models mentioned elsewhere in the thread).

Basically, 1 + 1 = 2 by definition. "1", "2", "+", and even "=" have no meaning by themselves. They're slots in the structure, and one of the properties of the structure is that "1 + 1 = 2". This means that for any model of the structure, you can apply the model's version of "+" to the model's version of "1" and itself, and the result with be equal (in the model's sense of "=") to the model's version of "2".

This fact can be proven from the Peano axioms, as other commenters have done, so any model which satisfies the Peano axioms satisfies this property.

So sure, you can have a situation where 1 + 1 does not equal 2, but then you're not talking about an instance of the natural numbers structure.
#12
04-18-2006, 11:13 AM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 12,834
Quote:
 Originally Posted by Cervaise That's nothing. Can you prove that .999... + .999... = 2?
Now that's being naughty, because in that equation "2" has to be a real number, not just a natural number. And you could ask, "Can you prove the 1/1 + 1/1 = 2" -- and then 2 becomes another kind of number, a rational number.

Of course, there's an obvious and natural isomorphism between a subset of the real numbers and the rational numbers, and another obvious and natural isomorphism between a subset of the rational numbers and the natural numbers. However, in the three different number systems, there are different axioms and definitions, and hence different proof methods.
#13
04-18-2006, 11:19 AM
 lieu Member Join Date: Aug 2001 Location: Bedrock Posts: 26,079
Quote:
 Originally Posted by Giles At another extreme, Alfred North Whitehead and Bertrand Russell's book Principia Mathematica notoriously says on p. 360: "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." That is, PM takes 360 pages building up an axiomatic system for the natural numbers, at the end of which you can say that you have proven that 1+1 = 2. (I've only skimmed PM, but I can assure you that those 360 pages are heavy reading indeed.)
Yep, that what I came in to reference. Don't I remember correctly too the after writing this Whitehead was never again the same mentally? I believe the inference I'd received was that proving 1 + 1 = 2 had in effect used him all up and broke his brain.
#14
04-18-2006, 12:26 PM
 MissMossie Guest Join Date: Feb 2004 Posts: 1,169
Quote:
 Originally Posted by Cervaise That's nothing. Can you prove that .999... + .999... = 2?
Better yet is 3/3=1? 1/3=.33333... and 2/3=.66666.... Now, assuming standard addition rules hold true, we know that 1/3+2/3=3/3, but .33333...+.66666... should equal .99999..., so does 3/3 really equal one, or does it just approach one?

This whole thread makes me think of a t-shirt one of my friends has. It says, "2+2=5, for exceptionally large values of 2."
#15
04-18-2006, 12:37 PM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 12,834
Quote:
 Originally Posted by MissMossie Better yet is 3/3=1? 1/3=.33333... and 2/3=.66666.... Now, assuming standard addition rules hold true, we know that 1/3+2/3=3/3, but .33333...+.66666... should equal .99999..., so does 3/3 really equal one, or does it just approach one?
That's a repeated subject of debate here: see 1 = 999... and .999~ -- and those threads have references to earlier threads on the subject.

(And the correct answer is that the infinitely long decimal fraction .999... is equal to 1).
#16
04-18-2006, 12:46 PM
 NurseCarmen Member Join Date: Mar 2002 Location: The Zen Arcade Posts: 8,288
1+1=4 for very large values of 1.
#17
04-18-2006, 01:29 PM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 12,834
Quote:
 Originally Posted by NurseCarmen 1+1=4 for very large values of 1.
More seriously,
1+1 = 10 in binary arithmetic, i.e., arithmetic base 2
1+1 = 0 in arithmetic modulo 2
#18
04-18-2006, 01:33 PM
 Exapno Mapcase Charter Member Join Date: Mar 2002 Location: NY but not NYC Posts: 29,348
1 + 1 = 1, for cases of love.
#19
04-18-2006, 02:29 PM
 nivlac Charter Member Join Date: May 2001 Location: Golden State Posts: 2,356
Quote:
 Originally Posted by Cervaise That's nothing. Can you prove that .999... + .999... = 2?
Yep. This is just a corollary of the theorems that .999... = 1 and 1 + 1 = 2. Q.E.D.
#20
04-18-2006, 02:52 PM
 RickJay Charter Jays Fan Moderator Join Date: Jun 2000 Location: Oakville, Canada Posts: 38,465
Quote:
 Originally Posted by Exapno Mapcase 1 + 1 = 1, for cases of love.
Forget the birth control, and 1+1=3.
#21
04-18-2006, 03:03 PM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 73,022
Quote:
 This is going to move to GD fairly quickly. Basically you're asking if there are un-provable statements within mathematics that make it inconsistent. If math is inconsistent then anything that uses mathematics to model empirical data must also be inconsistent. The thing is there are “unprovable” statements in mathematics called axioms. It's from these axioms that the resulting structures hang. You can have a variety of mathematical structures, of course, all based on differing starting axioms. The thing is, within those structures the mathematical reasoning is consistent, even though there maybe statement not provable within that framework.
Except that axioms are provable. The proof of an axiom consists of the statement of the axiom, and commenting that it is, in fact, an axiom.

There are, however, statements which cannot be proven. Gödel famously proved that in any self-consistent system sophisticated enough to encompass arithmetic, there must exist statements which are true, and yet which cannot be proven within that system. For instance, it is possible to express in the notation of such a system a statement G equivalent to "This statement cannot be proven within this system". If statement G is true, then by its own truth, we can deduce that G cannot be proven. If, on the other hand, G is false, then it cannot be proven by virtue of being false, since a self-consistent system cannot prove a falsehood. Therefore, either way G cannot be proven. But that's exactly what G is saying: G says that it can't be proven, and in fact, it can't be. So G is true. Ergo, there exists at least one statement in any given system which is true, but which cannot be proven in that system.

Now, that one's a pretty pathological case, and aside from exposing a quirk of mathematics, it doesn't really do much for us. But there are more useful examples. For instance, in geometry, exactly one of the following statements is true:

The sum of the angles of a triangle is exactly 180 degrees
The sum of the angles of a triangle is less than 180 degrees
The sum of the angles of a triangle is greater than 180 degrees

One of those three must be true, and the other two must be false. Which one's the true one? There's no mathematical way to know. Unless you take one of those statements, or its equivalent, as an axiom, you can't prove it. You can, however, answer the question scientifically (or at least attempt to do so), by measuring the angles of real triangles.
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#22
04-18-2006, 03:49 PM
 Mathochist Member Join Date: Feb 2004 Location: New Orleans, LA Posts: 3,031
Quote:
 Originally Posted by MissMossie Better yet is 3/3=1? 1/3=.33333... and 2/3=.66666.... Now, assuming standard addition rules hold true, we know that 1/3+2/3=3/3, but .33333...+.66666... should equal .99999..., so does 3/3 really equal one, or does it just approach one?
You're omitting things here. If you want to use ".9999... approaches 1", then you also have to use ".3333... approaches 1/3". If you want to use "1/3 =.3333...", then you also have to use "1 = .9999..."
#23
04-18-2006, 04:05 PM
 Cervaise Guest Join Date: Mar 1999 Location: underpants Posts: 19,743
Quote:
 Originally Posted by Giles Now that's being naughty
—bats eyes innocently—
#24
04-18-2006, 04:12 PM
 ultrafilter Guest Join Date: May 2001 Location: In another castle Posts: 18,988
Quote:
 Originally Posted by Giles Now that's being naughty, because in that equation "2" has to be a real number, not just a natural number. And you could ask, "Can you prove the 1/1 + 1/1 = 2" -- and then 2 becomes another kind of number, a rational number. Of course, there's an obvious and natural isomorphism between a subset of the real numbers and the rational numbers, and another obvious and natural isomorphism between a subset of the rational numbers and the natural numbers. However, in the three different number systems, there are different axioms and definitions, and hence different proof methods.
I got corrected on this before, so let me share the wisdom with you: in the standard construction, N and Q and R are all different types, but there's no reason why you couldn't construct R out of whole cloth and still get all the right properties. So in that case, that all becomes a non-issue.
#25
04-18-2006, 04:33 PM
 ultrafilter Guest Join Date: May 2001 Location: In another castle Posts: 18,988
Quote:
 Originally Posted by Giles Now that's being naughty, because in that equation "2" has to be a real number, not just a natural number. And you could ask, "Can you prove the 1/1 + 1/1 = 2" -- and then 2 becomes another kind of number, a rational number. Of course, there's an obvious and natural isomorphism between a subset of the real numbers and the rational numbers, and another obvious and natural isomorphism between a subset of the rational numbers and the natural numbers. However, in the three different number systems, there are different axioms and definitions, and hence different proof methods.
I got corrected on this before, so let me share the wisdom with you: in the standard construction, N and Q and R are all different types, but there's no reason why you couldn't construct R out of whole cloth and still get all the right properties. So in that case, that all becomes a non-issue.
#26
04-18-2006, 04:55 PM
 J String Guest Join Date: Jul 1999 Posts: 376
Quote:
 1+1 = 10 in binary arithmetic, i.e., arithmetic base 2
thus: "There are 10 kinds of people in the world; those who understand binary and those who don't."

(Man, I gotta find a t-shirt with that on it...)
#27
04-18-2006, 05:10 PM
 sverresverre Guest Join Date: Apr 2006 Posts: 3
My question was a simple one, and I think that most of the answers handled it in a brilliant way, but to some respect none of you handled it. Or maybe I am looking for something that is not there. It feels like that soar on the top of your mouth that would heal if you could stop tounging it, but you can't.

Math has spawned most of the technical achivements in history, but that does not prove anything.

The reason why I don't really think maths is the right way to go to prove this, at first glanze silly question, is that it is the very basis of maths. The only one who came close to any kind of answer was Gilles and Bathas, since they understood my gut-feeling approach to the problem. I have done more maths than average people, but I feel some kind of fundamental problem with 1+1=2. Why?

I think that no one can please me, but I am so glad so many tried. The few halfwits that tried to joke it away I understand, and I can't think of any other place were I would pose such a question, than here. I guess I wish there was a word or sentence so elegant that it would sum it all up. But I guess that it is not possible to be more elegant than:

1+1=2
#28
04-18-2006, 05:20 PM
 El Perro Fumando Guest Join Date: Apr 2006 Location: DC Area Posts: 384
Quote:
 Originally Posted by J String thus: "There are 10 kinds of people in the world; those who understand binary and those who don't." (Man, I gotta find a t-shirt with that on it...)
All you have to do is ask
#29
04-18-2006, 05:21 PM
 ultrafilter Guest Join Date: May 2001 Location: In another castle Posts: 18,988
Quote:
 Originally Posted by sverresverre ...it is the very basis of maths.
It's not. The basis of math is either set theory or category theory, depending on who you talk to. Everything else is defined in terms of those objects.

It may be the first thing you learn, but don't confuse that with it being fundamental. As Russell and Whitehead showed, arithmetic is actually pretty high-level stuff.
#30
04-18-2006, 05:38 PM
 chrisk Charter Member Join Date: Nov 2003 Location: Southern ontario Posts: 6,568
Quote:
 Originally Posted by sverresverre The only one who came close to any kind of answer was Gilles and Bathas, since they understood my gut-feeling approach to the problem. I have done more maths than average people, but I feel some kind of fundamental problem with 1+1=2. Why?
- I'm slightly confused... it doesn't look like anyone called Bathas has posted or been referred to in this thread.

- I'm sorry that those of us who might have taken the more technical approach weren't able to address your intuitive discomfort, but I can't think of anything more to say if you can't explain your problem a bit more clearly.

- Hi, Opal! (Yes, I know I'm cheating. I don't care.)

- I agree with those who said that math is by its nature abstract, with reasoning and axioms being the only true test of what is 'proven' within the system. Math is useful in modelling the real world, but that in and of itself doesn't make an equation true or false. It just makes it useful
#31
04-18-2006, 05:41 PM
 Ludovic Charter Member Join Date: Jul 2000 Location: America's Wing Posts: 27,181
Quote:
 Originally Posted by chrisk - I'm slightly confused... it doesn't look like anyone called Bathas has posted or been referred to in this thread. - I'm sorry that those of us who might have taken the more technical approach weren't able to address your intuitive discomfort, but I can't think of anything more to say if you can't explain your problem a bit more clearly. - Hi, Opal! (Yes, I know I'm cheating. I don't care.) - I agree with those who said that math is by its nature abstract, with reasoning and axioms being the only true test of what is 'proven' within the system. Math is useful in modelling the real world, but that in and of itself doesn't make an equation true or false. It just makes it useful
In this case, 1 + 1 + 1 = 4
#32
04-18-2006, 05:45 PM
 sverresverre Guest Join Date: Apr 2006 Posts: 3
I must have misread a nickname, so it's whoever is closer to that then

I can't formulate a proper question, because what I looked for was discourse, and in this there might be some kind of answer. But I think if you read the last sentence of my last post, it sort of struck me what the real issue was. Some things are better left unexplained, since they are inheriently beautiful and elengant. Or something like that. G'nite
#33
04-18-2006, 05:47 PM
 Cabbage Guest Join Date: Sep 1999 Location: Radford, VA Posts: 2,387
Quote:
 Originally Posted by sverresverre My question was a simple one, and I think that most of the answers handled it in a brilliant way, but to some respect none of you handled it. Or maybe I am looking for something that is not there. It feels like that soar on the top of your mouth that would heal if you could stop tounging it, but you can't. Math has spawned most of the technical achivements in history, but that does not prove anything. The reason why I don't really think maths is the right way to go to prove this, at first glanze silly question, is that it is the very basis of maths. The only one who came close to any kind of answer was Gilles and Bathas, since they understood my gut-feeling approach to the problem. I have done more maths than average people, but I feel some kind of fundamental problem with 1+1=2. Why? I think that no one can please me, but I am so glad so many tried. The few halfwits that tried to joke it away I understand, and I can't think of any other place were I would pose such a question, than here. I guess I wish there was a word or sentence so elegant that it would sum it all up. But I guess that it is not possible to be more elegant than: 1+1=2
I'm not sure you're really being quite clear on what you're looking for. This is my first post in this thread, and I'll admit I haven't read it in detail. I did read Giles post, since you referenced it (I don't know who you're referring to as "Bathas"). At any rate, I agree with Giles:

To ask for "a proof that 1+1=2" is meaningless until you've defined the semantic content of the symbols, "1+1=2".

There are various correct ways of defining this (and by "correct" I mean they are all both logically consistent, for one thing, and that they coincide with our intuitive notion of what "1+1=2" means).

Russell and Whitehead did it one way, I'm sure others have done it other ways.

If you're thinking in terms of the history of math, I'll first give a caveat that I'm no expert in the history of math, but I also feel pretty confident in saying that the original definition "1+1=2" was an intuitive one, and that it's true by definition.

The individual symbols, "1", "2", "+", and "=" were taken to express (intuitively) two different quantities, an operation of "accumulating" (for lack of a better description off the top of my head), and an idea of "sameness", respectively. I'm being very vague here, I know, but historically the foundations of math are intuitive and vague (by modern standards of rigor in math).

Given these intuitive definitions, 1+1=2 is a self evident truth: "2" is defined to represent the quantity (i.e., is "the same as", i.e., "=") when a single object (whose quantity is represented by "1") "accumulates" ("+") with another single object ("1").

In a modern, rigorous context, we need formal definitions of these symbols. As I mentioned, there are various ways of doing this so that we may then formally prove that 1+1=2.
#34
04-18-2006, 06:02 PM
 foolsguinea Charter Member Join Date: Jan 2000 Location: Tornado Alley Posts: 14,754
Quote:
 Originally Posted by Oy! Now, the interesting thing about this totally artificial construct mathematics is how useful it is in describing reality. We have never seen a situation that could be represented by 1 + 1 that did not equal 2. We don't expect we ever will.
Nitpick: With discrete objects & natural numbers, this is true. With estimated quantities, not so much. I'm not remembering the formal way to put this, but...
1.0 + 1.0 = 2.0 with a possible deviation of .099 = 1.90 to 2.10 exclusive.
1.3 + 1.4 = 2.7 with a possible deviation of .099 = 2.60 to 2.80 exclusive.
0.6 + 0.9 = 1.5 with a possible deviation of .099 = 1.40 to 1.60 exclusive.
1 + 1 = 2 with a possible deviation of .99 = 1 to 3 exclusive.

So, with discrete units, 1 + 1 = 2 by definition, but with imprecise quantities, 1 + 1 = 2 on average.
#35
04-19-2006, 12:03 AM
 Derleth Guest Join Date: Apr 2000 Location: Missoula, Montana, USA Posts: 19,825
sverresverre: You are approaching the entire field of mathematics with the wrong assumptions, and you are wondering why you can't get the answers you want to your questions. Your question has been answered just fine multiple times, but your reactions to those answers indicate to me that you aren't getting something very fundamental about the distinction between mathematics and science.

Try this on for size: Mathematics is just a game and in a game, things mean what we want them to mean. We can prove things in mathematics by shuffling symbols around in predefined ways because we have defined the concept of proof and all of those symbol-shuffling techniques beforehand, and we can agree on the fundamentals of the symbol-shuffling game. There is no reason why mathematics has to match up with anything whatsoever in the real world. It is, after all, just a game.

Except it does. We've been jiggering the rules of the game long enough we've come up with certain ways of playing the game that yield useful predictions about how the real world will behave in certain situations. This is so effective that great minds have pondered why it should be so unreasonably effective. This is a paper on the unreasonable effectiveness of mathematics in the natural sciences. I think your skepticism comes from the fact that we've made up this game and somehow get to apply it to the real world, and that it's somehow very deeply intertwined with science to the point where you cannot reasonably discuss fields like particle physics without throwing around some pretty serious equations. That unreasonable effectiveness is why: Mathematics has graduated from being a game to being a language, and it is a damned useful one in all of the hard sciences. (In fact, a 'hard' science is one where the predictions and results can be expressed numerically and as equations.)
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If you don't stop to analyze the snot spray, you are missing that which is best in life. - Miller
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#36
04-19-2006, 12:29 AM
 panache45 Member Join Date: Oct 2000 Location: NE Ohio (the 'burbs) Posts: 39,362
A question related to .99999...

I'm not even sure how to ask this question, but I'll try.

I accept the fact that .999...=1. Which means, the numbers in the series ".9, .99, .999, ..." are all <1, but gradually approach 1, and ultimately (at infinity) =1.

But there's another series of numbers, "1.1, 1.01, 1.001, ..." which are all >1, but gradually approach 1, and ultimately (at infinity) =1.

If the <1 series is symbolized by the number ".999...", how do we symbolize the >1 series? Somehow, "1.000..." doesn't seem right. My brain is trying to put a "1" at the end, but obviously that's not right either.

Help.
#37
04-19-2006, 12:30 AM
 panache45 Member Join Date: Oct 2000 Location: NE Ohio (the 'burbs) Posts: 39,362
Oops, that was supposed to be a new thread.

I'm not quite that insane. Yet.
#38
04-19-2006, 12:45 AM
 Cabbage Guest Join Date: Sep 1999 Location: Radford, VA Posts: 2,387
panache45, probably the simplest way to express such a series would be simply as:

lim 1 + 10-n, where the limit is as n goes to infinity.

We sometimes write "0.999..." to denote the limit of the .9, .99,... sequence, but there is no analogous way of writing this sequence.

The only way to write the limit as a decimal is as 1 or 0.999...; these are the only two decimal representations of 1. There is no decimal representation "1.000...1" (with a 1 after infinitely many zeros), it wouldn't make sense. There is no "omegath" digit in a decimal expansion (for simplicity, "omegath" digit can be taken to mean a digit with infinitely many digits preceding it)
#39
04-19-2006, 02:10 AM
 Tyrrell McAllister Guest Join Date: Feb 2001 Location: x∈S↔x≠x Posts: 871
Quote:
 Originally Posted by panache45 But there's another series of numbers, "1.1, 1.01, 1.001, ..." which are all >1, but gradually approach 1, and ultimately (at infinity) =1. If the <1 series is symbolized by the number ".999...", how do we symbolize the >1 series? Somehow, "1.000..." doesn't seem right. My brain is trying to put a "1" at the end, but obviously that's not right either.
How about 2 - 0.999... ?
#40
04-19-2006, 09:31 AM
 Mathochist Member Join Date: Feb 2004 Location: New Orleans, LA Posts: 3,031
Quote:
 Originally Posted by sverresverre Math has spawned most of the technical achivements in history, but that does not prove anything.
Actually, I think I see your problem. Evidently I wasn't clear in my first post.

Mathematics doesn't say anything about the real world. Mathematics can go from the Peano axioms to "1+1=2". Mathematics then says that for any model satisfying the Peano axioms the model also satisfies "1+1=2". Mathematics doesn't say anything about real-world examples of such models.

If you mean this as a real, physical question, ask a physicist. If you mean to ask the (deeper, unanswered) question of why mathematics is so "unreasonably effective" at describing the physical world, ask a philosopher. Mathematics has nothing to say.
#41
04-19-2006, 09:40 AM
 Mathochist Member Join Date: Feb 2004 Location: New Orleans, LA Posts: 3,031
Quote:
 Originally Posted by Cabbage If you're thinking in terms of the history of math, I'll first give a caveat that I'm no expert in the history of math, but I also feel pretty confident in saying that the original definition "1+1=2" was an intuitive one, and that it's true by definition. The individual symbols, "1", "2", "+", and "=" were taken to express (intuitively) two different quantities, an operation of "accumulating" (for lack of a better description off the top of my head), and an idea of "sameness", respectively. I'm being very vague here, I know, but historically the foundations of math are intuitive and vague (by modern standards of rigor in math). Given these intuitive definitions, 1+1=2 is a self evident truth: "2" is defined to represent the quantity (i.e., is "the same as", i.e., "=") when a single object (whose quantity is represented by "1") "accumulates" ("+") with another single object ("1"). In a modern, rigorous context, we need formal definitions of these symbols. As I mentioned, there are various ways of doing this so that we may then formally prove that 1+1=2.
Well, we could follow the categorifiers.

This is a relatively new push that's based on the idea that we shouldn't talk about identity, but rather only about isomorphism. Equivalence is not Equality. Confusing the two leads to decategorification, and most of mathematics is decategorified.

The natural numbers were "invented" to simplify (decategorify) the category of finite sets. To be exactly accurate here, I'm talking about cardinal numbers, since we identify sets with their cardinalities. Two sets with the same number of elements have the same cardinality. Given sets A and B with cardinalities a and b, the natural-numbers statement "a=b" really means "there exists a bijection between A and B". Addition is disjoint union of sets, while multiplication is Cartesian product.

So, within this context: prove that the disjoint union of two sets with one element each is in bijective correspondance with any set of two elements.

Now it comes down to set theory again.
#42
04-19-2006, 03:47 PM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 73,022
Quote:
 If the <1 series is symbolized by the number ".999...", how do we symbolize the >1 series?
The limits of both of those sequences are symbolized by the number "one", which may be written in two different ways in our number system, as either "1", or as ".999...".
#43
04-19-2006, 05:09 PM
 CurtC Guest Join Date: Dec 1999 Location: Texas Posts: 6,392
So for you math philosophers - recently I heard a creationist talking, and he said that the laws of logic are evidence of a creator. This got me to thinking that the laws of logic are the same kind of thing as the laws of mathematics. These creationist guys say that our beautiful and "unreasonably effective" mathematics are the way they are because they were defined by God. Which leads to the observation that if God wanted to make them some other way, I suppose like a universe where 1+1=3, he could damn well do that.

Other thoughtful people, when asked the silly question about whether God could make a rock so heavy that he could not lift it, point out that an omnipotent God could do anything that wasn't logically contradictory, such as a square circle is a contradiction. So I think these people are saying that the laws of logic (and math) would predate even God and he himself would be bound by them.

So is there any good observations about which would be right for the theist? I'd like a convenient way of refuting this argument for the existence of God, and this one seems pretty hollow to me, but I had not come across it before and was at a loss. What do you think?
#44
04-19-2006, 05:21 PM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 12,834
Quote:
 Originally Posted by CurtC So for you math philosophers - recently I heard a creationist talking, and he said that the laws of logic are evidence of a creator. This got me to thinking that the laws of logic are the same kind of thing as the laws of mathematics.
Yes, the laws of logic are the same as those of mathematics -- or perhaps better would be to say that mathematics is built on the laws of logic. However, you can build structures other than the natural numbers on the laws of logic: there are all kinds of algebra and geometry out there.

I'm not a professional philosopher, but my take on it is that the laws of logic and of mathematics are true for any possible universe, and they are true regardless of whether there is a creator. (At least in part that's because a creator would have to follow the laws of logic).

In any case, evidence of a creator is not the same as evidence of the Jewish/Christian God in the first few chapters of Genesis.
#45
04-19-2006, 05:51 PM
 Ludovic Charter Member Join Date: Jul 2000 Location: America's Wing Posts: 27,181
Quote:
 Originally Posted by Giles I'm not a professional philosopher, but my take on it is that the laws of logic and of mathematics are true for any possible universe, and they are true regardless of whether there is a creator. (At least in part that's because a creator would have to follow the laws of logic).
My take is that the theorems and logical structures would still have the same logical truth values, but might be rendered meaningless. For instance, what if there were a universe where addition had no meaning? The logical system called mathematics would still be sound but not applicable, much like most random logical systems we can come up with don't have much practical meaning.
#46
04-19-2006, 10:34 PM
 Hari Seldon Member Join Date: Mar 2002 Location: Trantor Posts: 11,215
Quote:
 Originally Posted by Mathochist Well, we could follow the categorifiers. This is a relatively new push that's based on the idea that we shouldn't talk about identity, but rather only about isomorphism. Equivalence is not Equality. Confusing the two leads to decategorification, and most of mathematics is decategorified. The natural numbers were "invented" to simplify (decategorify) the category of finite sets. To be exactly accurate here, I'm talking about cardinal numbers, since we identify sets with their cardinalities. Two sets with the same number of elements have the same cardinality. Given sets A and B with cardinalities a and b, the natural-numbers statement "a=b" really means "there exists a bijection between A and B". Addition is disjoint union of sets, while multiplication is Cartesian product. So, within this context: prove that the disjoint union of two sets with one element each is in bijective correspondance with any set of two elements. Now it comes down to set theory again.
No it doesn't because category theory can be created independent of set theory. It isn't usually, but it can be in the same way as set theory can be (has to be) created without set theory. So a category consists of objects and arrows in some pre-theoretical universe with the well-known properties. And a model for set theory is a particular kind of category that I could define but won't since it won't mean much to most of the people reading it.

All this doesn't really get to the original question. Is math based on faith, just like religion. My answer is tentatively yes, but with a major difference. Math is tested all the time and passes all tests. Religious faith makes no testable predictions and so remains purely a matter of faith. When tests are attempted, such as the recent test of prayer, religious leaders insist that religion makes no such promises. I do think that some religions do and some don't. At any rate, may god strike me dead on the spot if he wishes to demonstrate his power.

Incidentally, a colleague of mine claims that 2 is defined as the cardinality of a pair of platinum-iridium balls held in a termparature controlled vault in Sevres, France.

Ultimately, as many have said, give me your definition of 1, 2, +, and = and, i they are at all familiar ones, I can give you a proof. For a category theorist, 1 is a terminal object and 2 an object of truth values in a certain topos model of set theory and that fact that 1 + 1 = 2 is implicit in the model. Regardless, it makes the prediction that one ball taken with another ball gives two balls and that is verified regularly in every-day life, so it works.
#47
04-19-2006, 11:07 PM
 CurtC Guest Join Date: Dec 1999 Location: Texas Posts: 6,392
Quote:
 Originally Posted by Hari Seldon Incidentally, a colleague of mine claims that 2 is defined as the cardinality of a pair of platinum-iridium balls held in a termparature controlled vault in Sevres, France.
That's great - I gotta remember that. But... he's not serious, right? Right?
#48
04-19-2006, 11:10 PM
 ultrafilter Guest Join Date: May 2001 Location: In another castle Posts: 18,988
Quote:
 Originally Posted by CurtC That's great - I gotta remember that. But... he's not serious, right? Right?
#49
04-19-2006, 11:26 PM
 Speaker for the Dead Guest Join Date: Jun 2000 Location: New Brunswick Posts: 4,404
Quote:
 Originally Posted by CurtC That's great - I gotta remember that. But... he's not serious, right? Right?
I think it's a joke on the Metre des Archives and the Kilogramme des Archives, which are platinum objects which define (or used to define) the length of a meter and the weight of a kilogram.
#50
04-20-2006, 03:38 AM
 Derleth Guest Join Date: Apr 2000 Location: Missoula, Montana, USA Posts: 19,825
Quote:
 Originally Posted by Hari Seldon All this doesn't really get to the original question. Is math based on faith, just like religion. My answer is tentatively yes, but with a major difference. Math is tested all the time and passes all tests. Religious faith makes no testable predictions and so remains purely a matter of faith. When tests are attempted, such as the recent test of prayer, religious leaders insist that religion makes no such promises. I do think that some religions do and some don't. At any rate, may god strike me dead on the spot if he wishes to demonstrate his power.
I have recently thought of the same thing, and came to a different conclusion: Math is not based on faith because any axiomatic system can and will be abandoned if it can be proven inconsistent. If an axiomatic system were inconsistent—if it could be used to prove both a statement and that statement's negation—it would be abandoned instantly because all proofs derived from it would be immediately suspect. Religion simply doesn't work that way.

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