In this thread, Liberal provided this link to a “proof” of “1 + 1 = 2”
From that website we have
Definition of addition: Let a and b be in N. If b = 1, then define a + b = a'
Definition of number 2: 2 = 1'
So, 1 + 1 = 1’ (By definition of addition with a = 1)
= 2 (By definition of the number 2)
To me, this does not constitute a proof. It is a simple and trivial substitution based on the definitions of 1, 2 and the ‘+’ operator.
Something like this is what I would consider a proof.
I guess the question for debate is, does any combination of axioms and definitions constitute a proof, or does there need to be some level of “non-obviousness” to the derivation for it to be counted as proof (and not just a simple matter of definition)
If any combination of axioms and definitions constitues a proof, I could say:
Theorem: 1 + 1 = 1’
Proof: See definition of addition using a = 1.
To me, a proof has to be something more complex than just citing a definition. Am I wrong?
On the IBM-sponsored History of Mathematics wall (It’s in the Boston Museum of SCience, the Chicago Museum of Science and INdustry, and the L.A. Science Museyum, not to mention a poster thety put out) it claims that you have to go several hundred pages into Russwell and White’s book on Mathematical logic to get to the Proof of “1 + 1 = 2” (presumanly what you’re citing). I haven’t studied such rarefied math, but my understansding is that this involves some odd set theory, but looks an awful lot like a definition to me, too. It’s always seemed to my admittedly untutored soul that you could define
1 + 1 = 2**,
as well as **
1 + 2 = 3
1 + 3 = 4**, at which point, having defined the terms, you could finally “prove” that
2 + 2 = 4
With 1 + 2 = 3 you could prove that 2 + 1 = 3, but that seems like a proof of commutivity, rather than of addition. With 2 + 2 = 4 you’re finally performing an operation other than a definition to get the same result as defined by 1 + 3 = 4.
I’m sure a math expert will be along shortly to show us the error of our ways.
It is all going to depend on how you define “1”, “+”, “=” and “2”. It will also depend on what axioms you choose for your number system. (Axioms are propositions that are assumed withoutt proof). Hoowever, if you do not define “2” as “1 + 1”, then yyou ought to be able to prove the proposition “1 + 1 = 2”. If you can’t, then you don’t have a good model of the natural numbers and the addition operation
However, there are many diffferent models, and so many different proofs.
Many thanks for bringing this here, Polerius - that is very considerate and I’m extremely grateful.
There clearly are things which one must appeal to petitio principii, ie. circularly, but this is not quite one of those things: it comes from starting points defining ‘1’ and ‘truth’ and the like. Granted, it’s not much further along, but it is still not circular itself.
I’m a frayed knot. You would be imposing that mathematical sentence onto reality, not deducing it from reality. Me saying “the cat sat on the mat” doesn’t mean that the real cat really did sit there. Mathematics and, say, science, are different epistemologies.
I would disagree, SentientMeat. Contrary to Liberal’s assertions in the link thread, I would suggest that it is possible to prove “1+1=2” scientifically, in pretty much the manner that Scott_plaid suggests: by making repeated observations of what happens when you put one object next to one object.
This would not constitute a mathematical proof, but it would constitute a scientific one.
I’d say it’s more of a matter of definition of the symbols used.
I think it’s a bit disingenuous to claim Russell and Whitehead proved 1+1=2 in Principia Mathematica and just leave at that. What R and W were trying to do was axiomatize all of mathematics into a formal system of axioms and logic. In doing this, they had to rigorously define symbols such as 1, +, =, and 2. Of course, at this point, it was necessary to show that certain “obvious” statements were true, such as “1+1=2”. In this context, I think it’s more appropriate to claim that R and W were not so much proving that 1+1=2; they were really proving that their axiomatization made sense, and modelled our basic intuition of mathematics.
And we end up with … one object next to another. What if I said that was 3? If we are deducing 1+1=2 from reality, how would you prove me wrong? If you direct me to a dictionary or a maths book all you are doing, like I said, is imposing 1+1=2 onto reality.
The link Liberal provided to you, is indeed a link about the proof, that 1+1=2.
However, in this proof there are a lots of declarations (I count 5 axioms and 2 definitions) while the actual proof involves only a single step. So you can easily get the impression, that its just another definition. However: Try to prove 2 + 3 = 5.
This is a bizarre argument. Suppose I claim you can’t prove the existence of gravity scientifically. You show me numerous examples of objects acting under an approximately inverse-square attractive force proportional to the product of their masses. And I reply with “And we end up with … two objects next to each other. What if I said that was ‘up’? How would you prove me wrong?”
You’d laugh at me, and rightly so.
The reason you can’t point to two objects and claim that they’re three is because, speaking from a scientific standpoint, we have a common pool of observations and phenomena to which we can point and say “that is a set of two objectss, but that over there is a set of three.” Such definitions aren’t just pulled out of thin air, they’re modelled on the reality all around us and are as valid scientifically as our definition of mass. At the same time we can both perform experiments…namely, putting one object next to another…and observe the outcome of those experiments, and I confidently predict we’ll both see the same outcome, namely two objects.
If you stand there say “Oh yeah, well I say it’s three objects, prove me wrong”, and I prove you wrong by pointing to our common definition of what “three” means, I’m no more imposing addition onto reality than I’m imposing gravity onto reality by pointing out our common definition of mass.
Mathematics doesn’t use this epistemology much at all any more, but it’s silly to say that “1+1=2” can’t be proved scientifically. It can, just as much as the theory of gravity can be proved scientifically and in the same way: by repeated observations which have outcomes consistent with the outcome predicted by our hypothesis, using definitions of the terms involved that are modelled on previously observed phenomena.
Gravity being “the attraction between massive objects”. That definition sets up a falsifiable claim: finding no attraction between massive objects would falsify the statement “gravity exists”.
Again, how does one falsify 1+1=3, exactly?
Yet again, you are defining two and three rather than deducing them. You are deducing attraction between massive objects **from[b/] their behaviour.
Glad you agree that they are definitions. We agree that you cannot prove definitions scientifically?
Careful: it was gravity we were talking about just then, not mass. Scientifically proof of the existence of mass is a whole other pursuit.
But there’s no falsifiable consequence there. If I said “attraction” and you said “no attraction”, there are observable consequences which settle the matter. If I say “three!” and you say “two!”, you are merely asserting, definitionally.
Then I cordially invite you to my unbirthday party. More tea, Alice?
All analytic proofs have definitions. If they didn’t people would rightly complain that they don’t. There are also necessarily undefined terms (like “successor” in Peano’s axioms). There are then premises — i.e., assertions presumed to be true without proof. There follow inferences, the last of which is called “the conclusion”.
Science doesn’t prove anything true. It only proves things false. If it proved things true, then there couldn’t be new theories to displace old ones because once something is true — well, it is true. You can empirically test 1 + 1 = 2 all day long, and you have shown that you got two every time you tried. But you have not shown that you MUST get only two and nothing else every time you try in the future. Science doesn’t prove that gravity theory is true. *Equations * prove that gravity theory is true. Science, in failing to prove that gravity is false, is merely doing the job it is supposed to do.
If these matters — universally accepted (discounting crackpot revisionists) and perfectly sensible — are not conceded here, then… I don’t know. I just have no place here, I suppose.
Really, I believe that the conflict I see here is due to a conflict between mathematics being grounded by physics, and math being it’s own language. Then again, I can hardly understand the issue, at all. I just felt the below link was too good to waste.
Mathematics is not about science, in the sense that it does observations (=experiments) on phenomena (=measurements). It is strictly theoretical. But it is a very useful tool for science.
To understand the little sequence of characters “1+1=2” as a mathematical equation you need three things: 1. defintion of N (natural numbers, N, P1-P5), definition of how numbers are written if not (as in my above post) as follower of follower of … of 1 and 3. definition of addition ‘+’.
Given these definitions all you can prove is, that the addition of two numbers a,b element of N, results in a third number c that also is element of N. How you call them (one, two, three, …, or eins, zwei, drei, …, or 1, 1’, 1’’, … or 1, 2, 3, …) does not really matter, it is just a convention as is ‘+’ to symbolize addition.
Think about this number: (10^80)!
This is a very large number. You will never be able to write it down (there are only 10^80 atoms in the world and this number has more digits than that.)
So there is no way to ever prove scientifically, that this number is element of N. But it is easy to prove mathematically, after you have defined multiplication and faculty on N.
Now does this impose “(10^80)! element of N” on the reality? I doubt it. Numbers are nothing real, never were and never will be. That is why physicists use units like second, meter, kilogramm to work on their science.