So we’re getting into number representations…
Here’s a few:
1 + 1 = 11 (unary)
1 + 1 = 110 (base -2)
1 + 1 = 2 (Cantor expansion)
btw, Rhythmdvl, you are correct about the binary.
I was just messin around, Rhythm, with different ways of interpreting numbers.
One number plus another number equals only one number, not two separate numbers. 
–Tim
Now here’s some messing around: your statement is only true if we use the ordinary meanings of “plus” and “equals”. For instance, let a + b be defined to be {a, b}. Then the result of adding two numbers isn’t even a number.
Now, now - my brother has made an academic career out of worrying about this type of issue and Lord Russell’s work in general - surely you wouldn’t want to pull the bread from his children’s mouths?
Why the hell would anyone define 1 as anything other than 1 for anything other than “proving” 1 + 1 = 2?
It seems to me(and I am only 16) that many of you are struggling with the concept of a NUMBER and a NUMERAL.(and no I am not yelling I am just making them more noticable so someone doesn’t just pass over them). Now I don’t have a dictionary with me, but it seems to me that a NUMBER is an idea of quatity while a NUMERAL is a representation of that idea. Just because you call something a name doesn’t make it anything. And also, IMHO a NUMBER doesn’t change if you use a different base system, that is just a different representation of that same number, using a different means to find that NUMBER.
Just my 2 cents.
Thanks,
Matt Shepard
Well, we’re not really proving that 1 + 1 = 2. We’re proving that the Peano postulates and the definitions of addition and equality guarantee that 1 + 1 = 2. In order for this to be true, it has to hold for any possible meaning we give to the elementary ideas of “zero” and “successor” (because addition, 1, and 2 are defined in terms of those).
John Corrado’s post deals with fuzzy arithmetic, a different idea which is very useful for control systems and neural nets.
For the curious, the Peano postulates are as follows:
[list=1][li] 0 is a natural number.[/li][li] If x is a natural number, there is another natural number denoted x’ (called the successor of x).[/li][li] 0 is not the successor of any natural number x.[/li][li] If x’ = y’, then x = y.[/li] If Q is a property which may or may not hold for any given natural number then if 0 has property Q and x’ has property Q whenever x has property Q, then all natural numbers have property Q.[/list=1]
I too have a problem with this proof.
I’m skeptical that real world experiments can show anything about mathematics.
Let me give you an example: Suppose you decided to prove the Pythagorean Thereom by constructing a right triangle with “legs” of length 3 and 4, and measuring the hypotenuse to see if it is in fact length 5.
And suppose you measured the hypotenuese and discovered it was 5 1/2 units in length. IMHO this would not mean that the Pythagorean Thereom is incorrect. Instead, it is evidence that some other mathematical system (besides Euclidean Geometry) is a better model for the real world.
Similarly, if the hypotenuese turned out to be length 5, it wouldn’t “prove” the Pythagorean Thereom - it is merely empirical evidence that Euclidean geometry is an accurate model/representation of the real world.
So I would argue that your penny experiment is merely evidence that the standard rules of arithmetic are a useful or accurate model for real life.
Well of course we can’t PROVE anything. WE can only offer supportive evidence. We can prove something wrong, but we can’t truly prove anything because we cannot include EVERY variable possible.
At least that’s what I have gathered.
I vaguely remember reading on the SDMB over a year ago a conversation to this effect, and someone offering that “one drop of water plus one drop of water equals one drop of water.” Pennies don’t merge with each other when they touch; water droplets do.
Of course this doesn’t work when you measure water by mass or volume rather than “droplets”, but it’s still relevant to the topic.
[QUOTE]
*Originally posted by lucwarm *
**
So, if an engineer builds a bridge using Euclidean geometry and it collapses taking 200 commuters with it he should just say, “Well, I guess Euclidean geometry is only true on paper. Don’t blame me, blame Pythagoras.”
Of course real life tests should prove a mathematical or scientific concepts.
For example, lets look at the Avogadro Constant.
Lets say tommorow it is revealed that there’s actually 3.4 X 10 to the 9th atoms in one gram of hydrogen, not 6.02 X 10 to the 24th. And at standard temperature and pressure, in the gaseous state, that many atoms take up 22.4 liters.
Should we keep saying that a mole is 6.02 X 10 to the 24th? No! If a real life test disproves the abstract, it is the abstract which must change or be rejected.
oops I was quoting lucwarm above, not myself.
Yes and no. This is getting to be kinda GD, but real-life tests can only disprove a scientific concept. No one single event can prove such a thing; rather, a large number of events that fail to disprove it are taken as evidence that the concept is correct. But one event could still disprove it.
Mathematical concepts are a bit different. If a mathematician rigorously proves that “if A, then B”, then A will always imply B (assuming there are no errors or oversights in the proof). The only question is to whether A is a valid scientific model, and that goes back to the scenario above.
Your example of the engineer isn’t that good. Euclidean geometry has been a good model for local space ever since it was invented, so the fault probably lies with the engineer.
Qwertyasdfg, you might find the history of non-Euclidean geometry interesting, or at least relevant to this discussion.
All of a sudden, in the early nineteenth century, Lobachevski (among others) suddenly came up with a new, self-consistent, and theoretically rich branch of geometry that modelled nothing at all in the real world as far as they knew. (It is connected to relativity among other things, but that came later.) It marked the beginning of a paradigm shift in mathematics, if you’ll pardon the phrase. Before non-Euclidean geometry, math was very much like physics in that it was an attempt to describe real-world phenomenon, so the “penny proof” that 1+1=2 was all that anyone ever needed, and any theorem that contradicted 1+1=2 would have been dismissed out of hand. Nowadays, by contrast, a field of mathematics doesn’t have to model anything in the real world to be considered “good” (although it helps).
This is in stark contrast to, say, physics, which is why your analogy to the number of atoms in a mole doesn’t really apply.
It’s also worth mentioning at this point that there are multiple self-consistent theories of arithmetic, just as there are multiple self-consistent theories of geometry. In fact, there are an infinite number of such theories; Kurt Godel showed that earlier this century. None of those theories disagree on anything as mundane as 1+1=2, but they do disagree…
[nitpick]
Especially since Avogadros number is 6.02 X 10 to the twenty third !
[/nitpick]
it’s done in science all the time, to express the limit of accuracy of your measurements. for instance, if you have a measuring cylinder which is marked in 1 mL increments, a measurement of (say) 5mL means that you have an amount of liquid of between 4.5mL and 5.5mL.
Silly me! Its summer and my forgetting everything I learned has started to kick in.
Thanks, Mathgeek, I’ve been wondering about this for some time.
I’ve never even looked at Principia Mathematica (although I have a Math Ph.D. friend who has), but the statement that they “proved” that 1 + 1 = 2 has always bothered me for a fundamental reason – how do you then define 1 and 2? It seemed to me that 1 + 1 = 2 was itself a definition. I could then use 1 + 2 = 3 to define “3” , but I still wasn’t up to something I could prove (unless you want to prove commutivity by showing 1 + 2 = 2 + 1, something you can’t do with 1 + 1). It wasn’t until you got to 1 + 3 = 4 that you got to a point where you could break away from definitions and, finally, prove that 2 + 2 = 4 as well.