Some posts in this thread:
http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=152895
got me wondering. Does the fact that adding 1 apple to a pile of 1 apples give you a pile of 2 apples mean that 1+1=2, or does the immutable fact that 1 + 1 = 2 (or 10, or II, etc) cause you to get two appples in the pile? Are the above facts independant of each other?
No, maths is not independant of reality otherwise it would have no application in the real world. It is generally said that even the most and obscure and abstract branches of maths are eventually found some practical application.
If their truths are immutable, it is only because we won’t let them be anything else.
We define the system such that it reflects reality. I can design operations that don’t seem to correspond to any reality. 1+1=2 because we say that is what it means. We said that because it is damn convenient to do so as it does reflect what we see. Down the road a bit we also define 0! =1 . I forget why, but I never quite understood how that corresponded to reality in any way. There are systems where (a+b)^2 = (a^2)+(b^2) even though that is not true in the math we normally use. Euclidean geometry doesn’t reflect our best measurements of reality, but it is still used because it is useful and a pretty good approximation.
In pure mathematics, theories and ideas are either interesting or uninteresting. There is no correlation between the ideas and the real world, necessarily, or nothing could ever get done. You just start with a few assumptions and play along from there. The end product might have nothing whatsoever to do with the real world, or it might uncannily resemble it in some ways.
If you are building a bridge, on the other hand, your math has quite a bit to do with reality – or at least it should.
Theoretic math tries to develop a system that reflects reality, but it is not the same as reality. It is a model.
For obvious examples, a line in mathematics has no width; such a thing cannot exist in nature. A line in mathematics is infinitely subdivisible; again, in nature, you get so far into the subatomic level and you can’t really subdivide further.
The concept that one apple plus one apple gives two apples led to the mathematical model of arithmetic, where 1 + 1 = 2. But that’s not always true in the real world: scrambling one egg with one other egg gives you one omlette. Or one sperm plus one egg gives one embryo. Or adding one match to one cup of gasoline only gives one explosion.
no, because reality is socially constructed, so somewhere along the line, someone had to define that 1+1 = 2. otherwise, 1+1 could = 3 just because a long time ago someone made it so.
Math was inspired by reality, and it’s a very useful model.
However, some mathematical ideas have no real world analogues when they are invented (category theory, for instance).
At bottom, all that matters to many mathematicians is what’s interesting. What relation it has to the real world is irrelevant.
Hmmm. Actually 1 + 1 = 2 comes from counting. If there are this || many objcts, then we put them in one to one correspondence with our numbering system and there is one object identified with our number 1 and another our our number 2. It is the numbering system that is socially constructed, not the reality of || objects, or ideas, or whatever.
this is a question that has also been plaguing me of late.
my thoughts were more along the lines of logic than all of pure and applied mathematics. so let’s have a look:
is P -> P true, independent of any system? it certainly seems true independent of any real system. but we don’t know of any systems except for reality. if there can be a reality without logic (e.g. “the lord works in mysterious ways”), what can we say about it, meaningfully? does (can) it follow any of the fundamental ideas of logic (truth, the proposition, implication, etc.)?
can anyone make a case for or against a world that exists independent of logic? is there some reality where someone can draw a circle with 4 right angles?
Yep. Under the Manhattan metric, squares and circles are the same thing. The Manhattan metric is d((x[sub]1[/sub], y[sub]1[/sub]), (x[sub]2[/sub], y[sub]2[/sub])) = |x[sub]1[/sub] - x[sub]2[/sub]| + |y[sub]1[/sub] - y[sub]2[/sub]|.
But I suspect that’s only tangential to your question.
Just a note… It was said that 0! = 1, but no reason was given.
The reason is to keep up a pattern:
n!/(n-1)! = n
For example:
6!/5! = 6
5!/4! = 5
4!/3! = 4
3!/2! = 3
2!/1! = 2
1!/0! = 1
To make the last statement work, 0! has been assigned a value of 1, since 1/1 = 1. If that were not the case, then the equation would be 1/0, which is undefined. So the 0! is just to keep the model working, and it doesn’t break anything.