I always did better with multiple choice questions. What are our options?
Epimetheus: Yes, kind of. But that’s not exactly what I meant. I don’t simple mean changing the meaning of words here. I’m not that pedantic 
I mean, it depends what rules you define.
Example: Getting back to PTPQPP.
Axiom: PQP
Rules:[list=1]
[li]You may at any time add T to the left of Q[/li][li]You may add a P to the left of Q if you also add a P to the right of Q[/li][li]You may remove a P to the left of Q if you also remove a P to the right of Q[/li][/list=1]
Now we have a simple system. And, I said earlier that PTPPQPPP is true. Is it?
Start with:
PQP (axiom)
PPQPP (rule 2)
TPPQPP (rule 1)
PTPPQPPP (rule 2, and proved)
There, we have a simple logical proof for PTPPQPP. Does it have any basis in reality though? Well, if map P as a number, PP as the number twice (just added), T as add and Q as Equals, then yes we do.
In this sense T is redundant (PTP is the same as PP). Is it always redundant under the simple rules I’ve specified, or does it depend on the interpretation we put on it?
Anyway, we can say 2+2 = 4 with PPTPPQPPPP. Neat, huh? So it seems as though our little system here represents reality. And of course, 4 = 2+2 so PPPPQPPTPP.
Erm, no. It doesn’t. There is no rule allowing T added to the right hand side. You can’t derive PPPPQPPTPP from our axiom. So, our system doesn’t represent reality very well after all. Or maybe it does…after all, there are situations in reality where things can be added, but can’t be seperated. Mix two tins of paint for a simple example.
I hope this makes my point a bit clearer. 2+2 may or may not equal 4…
Mixing two tins of paint together is not the best analogy I think. Taking a gallon of white paint, and a gallon of blue paint and you get TWO gallons of light blue paint. You still have twice the amount, the contents just changed each other.
Seems to me mathmatics represents reality quite well, of course there are exceptions to it, takes a bit of a leap to say because of some exceptions (or strange conversions from numbers to letters and back again) that it does not represent reality very well.
I could be missing something though.
If 2+2 does not equal 4, dylan, then I am simply not sure what you mean by 2.
Yes, if we change definitions, statements no longer mean the same thing. How is this surprising or enlightening?
Well, the paint example does illustrate what I mean. 1 blue plus 1 white equals 2 light blue. But 2 light blue minus 1 white just can’t be done. But it works fine if you consider only volume…and that doesn’t always work…consider alcohol added to water.
So, maths represents reality only in certain idealised cases. In other words, it’s just a model, not the real thing.
I didn’t switch from numbers to letters to confuse the issue. It’s just that people are so familiar with the number symbols and +, -, etc that they forget that they are just symbols and manipulations of symbols.
You saw how my number system also represented (or could be mapped to) reality in the same way as 2+2=4 can. But it wasn’t a very good system, and so didn’t map very well. Now, I could have added a few more rules (allow T to be added to right of Q for example) to make it better. And, if I’d added a few more rules it would be pretty much the same as the arithmetic we’re familiar with.
But what’s being claimed is that the maths we have is reality. Well, I could equally well claim that the PQP system is real, and always real.
After all, Maud’Dib claimed:
Sure, I’ve used different symbols but so what?
Use logic to answer this question:
What should I eat for lunch today?
I don’t see how this matters one bit. What the 2 is, doesn’t matter. 2 quarts of water and 2 quarts of milk still equal 4 quarts, that does not change. You can separate those back into quarts. Because you cannot change them into milk and water again (without special tools), does not mean that 2+2=4 is invalid.
You have entered into the realm of semantic reasoning to disqualify mathmatics. Of course mathmatics is just a model to describe local reality in a particular circumstance. So is language, so is thought, so is your semantic reasoning. Has your semantic reasoning allowed things like the Eiffel tower to be built? I will stick to thinking mathmatics (and logic) is effective in describing my reality, due to what those “mere models” have allowed us to do.
Oh, you are adding units of different colors of paint. that is what you are adding, not the colors. (that is my point) How are you to quantify paint by seperate colors if you don’t use units like gallons, quarts, etc?
So is this sentence, whether written, gestured (if it could be gestured), or spoken.
Ahh, good. Yes, this is what I mean. What exactly do we mean by 2? I’m saying it’s a symbol we use to represent a concept, which may or may not have a basis in reality. Others are saying it’s real in and of itself. That’s what I’ve been trying (rather poorly, it seems) to get across.
But 90% of the problem of trying to explain it is that people are so familiar with these symbols they mistake them for real things. The other 10% is trying to remember not to take it for granted myself!
Epimetheus: Arghh! This is not semantics! Please try to understand what I’m saying rather than trying to pick faults in my analogies! I’m not trying to win an argument here, I’m trying to explain! The point of the alcohol to water example was that 1 quart water + 1 quart alcohol does NOT give you 2 quarts mix. Sorry if that was confusing, I’d badly assumed that this example was taught in most schools (stupid of me, since most of you aren’t even from the same country as me, never mind having the same curriculum!)
Of course Maths and Logic are useful models of reality…they allow really great things to be done. But they’re models. Not reality. And this is clearly illustrated by your choice of example. When does 2 + 2 = 4? In volume? Not always. In Fruit? Which fruit? It is a model that can be mapped to reality in a certain way.
Now, so could my PQP system, which is a mathematical system. What exactly makes yours more valid than mine?
erislover: Regarding sentences. No, they’re not. I have no set rules to tell me what I must write having written something previously. Nor to tell me that a sentence is correct or incorrect. Nor any axioms to work from. It’s totally different in fact.
So 2+2=4 is a model that does not depict reality because people think 2 of a reality. No, two is more than one of a certain thing. How we depict that certain thing is obviously dependent on what it is. 2 fruit is a reality, because we define an apple as one, two apples as 2, etc.
1 quart of water and 1 quart of alcohol does not equal 2 quarts mixxed is correct. If that was the conclusion you were looking for, then congradulations. However, it DOES equal 2 quarts. So unless you were specifically looking for alcohol and water to mix, then the model depicted reality quite well.
Hell, gravity is just a model, therefore gravity is not reality because it is just a model…not reality. :rolleyes:
I guess I am just dense, write me off then, but I will stick to thinking mathmatics is a working model that accurately helps describe reality. You can go on thinking it is not reality and just a model, just like evolution is a theory right?
I also thing language is a reality. It REALLY exists, so it is real. Of course by your seeming conclusions it is just a model as well, and does not actually exist.
Well, turn this sentence on itself, and one comes to the conclusion that people are so familiar with the qualifier ‘real things’ that they think they are “real things”.
Not a result you’d like to take very far, I’d imagine.
Hmm…seems I’m coming across as anti-science here. It’s not so! Anyway, it’s the “certain thing” part that makes it ambiguous. The universe isn’t as neat as maths.
The 1 quart water, 1 quart alcohol actually doesn’t add up to 2 quarts (try it and see!). That’s what I meant with that example.
Gravity is real. We have a Theory of Gravity and that is just the model. There is a difference between the two. Similarly, evolution certainly appears to be real and we have a Theory of Evolution. They are not the same thing (different theories might give different methods of selective advantage being passed on, for example).
No, you’re not being dense. Thanks for sticking with me actually; I’m not the best at explanations. And I haven’t enjoyed a discussion like this in a long time…
Hmm, weird. Is there an explination for why this occurs? Does water make the alcohol more dense, or vice versa? (guess it depends on if you use quart, or ounces as your unit of measure)
I can relate to that, I have problems explaining things as well as figuring them out. (I am the neanderthal with a club compared to the Olympic fencers on this board, me bashscratch head when I miss)
erislover: It’s more like I claim that my PQP system is an accurate representation of reality. Would you accept that claim?
Yup. It’s the molecular stucture. Because we’re talking volume here…think of it like adding a quart of sand to a quart of ping-pong balls…
Lest you think I am all alone in my view, I offer the following quotation from Bertrand Russell:
All that demonstrates to me is that I may not place the sign “2” in front of everything all the time.
Well, if they are dogs, or resemble dogs, you can’t compare them to a virus. (I know, you are just quoting him).
Basically he has broken down into semantical hashing so as to sound smarter and more "in’ with it. “When you have told me what you mean by entity, then we will resume the discussion” is a cop out, and intellectual tactic to try to muscle your way out of the argument, or if the argument continues, allows you the person to blather on about what this can be and what this cannot be. Totally unrelated to the argument. It is a diversionary tactic and that is all.