There is the notion of a set of elements.
{} is the set containing nothing. {bka, pha} is the set containing bka, and pha. A rigorous development is beyond the scope of a single post.
{{x},{x,y}} is the set containing {x} (which is in turn a set containing x) and {x,y} (which is in turn a set containing {x,y}).
A short form of writing {{x},{x,y}} is (x,y) and is called an ordered pair.
A set containing ordered pair is a binary relation. If R is a binary relation, {x|(x,y) in R} is called the domain of R and {y|(x,y) in R} is called the Range of a R. If for every x in the domain of R there is exactly one y in the range of R such that (x,y) is in R, then R is a '1-to-1' correspondence.
If the following three things are true of a relation of R, then R is an "equivalence relation":
1) (x,x) is in R
2) if (x,y) is in R, then (y,x) is in R
3) if (x,y) is in R and (y,x) is in R, then (x,z) is in R
The 0 is often short hand for {}. The 'successor' of a set, S, is the set containing S and the contents of S. The successor of 0 is {{}} also often known as 1. The successor of 1 is {{{}},{}} also often known as 2. You could continue on this way. The sets defined this way are the "whole numbers."
The number of objects in a set is defined to be the whole number to which you can create a 1-to-1 correspondence.
Now imagine that there is a set S, and binary relation, A1, consisting of ordered pairs of the form ((a,b),c), where a, b, and c are elements of S. Consider further the binary relation A2 which is {(c,(a,b))|((a,b),c) is in A1}, and yet another binary relation A3 consisting ordered pairs of the form (e,f) where e and f are from S. If the union of A1, A2, and A3 forms an equivalence relation, then A1 and A3 form a basis for an "addition" of sorts (consider modulo arithmetic for instance).
It is often customary to use the whole numbers as S, A1 with the following properties:
1) (({},{}),{}) is in A1
2) if ((a,b),c) is in A1, then ((successor of a, b),successor of c) is in A1
3) if ((a,b),c) is in A1, then ((a, successor of b),successor of c) is in A1
and A3 being {(x,x)| x is a whole number}
You can check that the appropriate equivalence relation holds.
You could then introduce '+' as notation so that a+b=c means ((a,b),c) is in A1.
Obviously, I can't make all this rigorous in one post. But hopefully it gives you some idea how mathematicians "built-up" addition conceptually from sets.
To unravel all this for 2+2:
We look for the appropriate element in the set of the form (({{{}},{}},{{{}},{}}),c) in A1.
Let us build up. (({},{}),{}) is in A1. So (({{}},{}),{{}}) is in A1. Which in turn means, (({{}},{{}}),{{{}},{}}) is in A1. This then means (({{{}},{}},{{}}),{{{{}},{}},{{}},{}}). Note {{{{}},{}},{{}},{}} is often referred as 3 and is the successor of 2 which is {{{}},{}}. Finally, we arrive at what we are looking for in that (({{{}},{}},{{{}},{}}),{{{{{}},{}},{{}},{}},{{{}}, {}},{{}},{}}) is in A1. {{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}} is often refered to as 4.
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
2+2=5
Did the black hole suck us in yet?
~I'm looking California, and feeling Minnesota.
Shameless Self-Promotion:MDP2525's Den and the Start of Motorcycle Maintenance
2 + 2 = 1
Two piles of sand plus two piles of sand equals one pile of sand.
(One must consider units to get a meaningful result.)
INTJ | 5w4 - Sp/Sx/So | 5-4-(9/1) | RLoEI | Melancholic-Choleric | Johari & Nohari
This will not end well...
But it will at least be poetic, I suppose...
Hmm... But what if it does end well?
Then I suppose it will be a different sort of poetry, a preferable sort...
A sort I could become accustomed to...
Formal reasoning is formal reasoning, whether you learn it in philosophy or in math. I believe math teaches it better, personally.
Actually not very long. I didn't try to make it rigorous, just show one possible mathematical definition for 2+2.
Most mathematicians and computer scientists who have studied sets could do a similar explanation in a few minutes.
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
Philosophical Logic
From wiki:
To the bolded. I dont study the relationship between logic and speech. I study the speech itself and the logistics within the speech. This is how I listen and understand. What I do goes beyond if X then Y, or If not A then not B. Its not mathematical logic. I dont study logic, I practice logic.Philosophical logic
Philosophical logic deals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the traditional discipline that was called "Logic" before the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g., free logics, tense logics) as well as various extensions of classical logic (e.g., modal logics), and non-standard semantics for such logics (e.g., Kripke's technique of supervaluations in the semantics of logic).
Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure his own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many people are untrained in logic and unaware of how to formulate an argument correctly.
Take what I say with a grain of salt, because that's all it is compared to the ocean of complexity when it comes to actions and real life.