In two separate discussions so far, I've seen people imply that maybe it isn't tautologically true that 2+2=4 or that 7+5=12. My interest piqued, I went googling. I found this article:
Geometry and Arithmetic are Synthetic - http://www.earlham.edu/~peters/writing/synth.htm (http://www.earlham.edu/~peters/writing/synth.htm)
If interested at all, but only in the topic I've raised, skip to the following paragraph and start reading.
Quote from: "The Article"Now we may turn to the second argument for the thesis that arithmetical truths are synthetic. For this argument we will state an arithmetical truth in the notation of modern predicate logic, and then test its analyticity or logical truth.
I dispute the fellow's logic. Primarily, I dispute that he set up his algebra correctly, when he wrote -
Quote from: "The Article"Equal(add(a,b),c).
That isn't what we see when we look at 2+2=4 or 7+5=12 - or so I would argue. I, for one, see this:
Equal(add(c-b,b),c).What I see is tautologically true, and would pass the
logical validity test presented in the article.
Does anyone have further insight into any of this?
The author (or perhaps Nelson) claims to be establishing a formula of some sort but I wouldn't hardly call it that. It's merely an array. He starts by giving the symbols real meaning but veers from it completely once there are no restrictions to what he decides to put into the "formula". I could've made the same argument by doing this:
2,2)4
5,7)12
111,200)100
Dropping random numbers in between symbols doesn't prove anything, but it is an interesting read. I think he does his own "formula" in when he says
QuoteSince it is easily shown that only logically valid formulas possess the property that their negations are contradictions, then only logically valid formulas are analytic. So the question whether arithmetical truths are analytic has become whether they are logically valid formulas. If they possess only one of the weaker grades of truth, then they are synthetic.
Mathematical formulas have to be proven to meet logical validity before accepted. He's trying to redefine math.
1 + 1 = 10
4 + 4 = 10
8 + 8 = 10
Are also all true and I use them all quite often. The values, numbers, rules of math and symbols are all used represent things, but they in themselves are useless and mean nothing.
In my examples at the start: the first is binary, the second is oct and the third is hex. While the same rules of math are applied to the different bases, my examples were meant to show that "2 + 2 = 4" is only true in some cases (only number systems with a base of 5 or higher). We didn't discover math as we did things like gravity and DNA, we (not really me included, but I like to egotistically include myself and all humans with all the great people that did do awesome things) created it.
I also wanted to demonstrate that the rules can act independently of the number base that one works in. An argument against this point is that 4 apples is 4 apples no matter how you represent the number, but that also means that one is accepting that math, the symbols and the number base are independent of real objects. Accepting that math is a man made concept and not a reality. A creation and not a discovery. That's my opinion anyway.
I can recall Descartes uses something like this to eventually "dispel" demon doubt and attempt to prove a gods existence.
It could be an issue of what you consider to be qualitatively identical and numerically identical.
Quote from: "Davin"Accepting that math is a man made concept and not a reality. A creation and not a discovery. That's my opinion anyway.
But math is like a language. Simply because what we use are representational figures wouldn't mean it's not analytical. Wouldn't that be like saying writing about any "discovery of DNA" would mean it was a man-made concept because when I write "DNA" it's not
actual DNA?
Quote from: "Sophus"Quote from: "Davin"Accepting that math is a man made concept and not a reality. A creation and not a discovery. That's my opinion anyway.
But math is like a language. Simply because what we use are representational figures wouldn't mean it's not analytical. Wouldn't that be like saying writing about any "discovery of DNA" would mean it was a man-made concept because when I write "DNA" it's not actual DNA?
I never said that it's not analytical. And the concept of DNA is not DNA itself. The difference is, no one can trip over "2 + 2 = 4" while tripping is evidence for gravity. "2 + 2 = 4" only exists as a concept, there is no relation to reality unless applied, while gravity is there whether we apply a concept to it or not.
Quote from: "Davin"Quote from: "Sophus"Quote from: "Davin"Accepting that math is a man made concept and not a reality. A creation and not a discovery. That's my opinion anyway.
But math is like a language. Simply because what we use are representational figures wouldn't mean it's not analytical. Wouldn't that be like saying writing about any "discovery of DNA" would mean it was a man-made concept because when I write "DNA" it's not actual DNA?
I never said that it's not analytical. And the concept of DNA is not DNA itself. The difference is, no one can trip over "2 + 2 = 4" while tripping is evidence for gravity. "2 + 2 = 4" only exists as a concept, there is no relation to reality unless applied, while gravity is there whether we apply a concept to it or not.
Oh, okay. I gotchya.
NEXT EQUATION: 7 x 13 = 28
[youtube:3oargis0]http://www.youtube.com/watch?v=yHZzObQUgE8[/youtube:3oargis0]
8 + 9 = 17 on my calculator, but 5 on my clock. Go figure.
Your clock uses Z12 equivalence classes starting at 1 instead of 0.
My clock uses Z24 starting at 0.