OK, that's fine. OF course just to make it more precise. I said almost all of mathematics before the era of set theory can be formalized as an extensi...
I didn't ask about those which CANNOT, I asked about those which are not. I mean are not presented in a formal manner as an extension of first order l...
Yes! On the informal level I would agree, but formally NO. It only symbolizes a number that the operator + is sending the pair (2,2) to. It doesn't sp...
Well PA is a mathematical system. Most formal mathematical systems nowadays are stipulated as extensions of logical systems, in particular first order...
possibly I don't agree with R&W on that. + is not a logical operator, it is a mathematical operator, but as you know we can speak logically about extr...
Here is your error, you think that '2 + 2' denotes TWO objects. This is wrong. You are not understanding the operator "+", this is a two place FUNCTIO...
You say '2+2' denotes an object divided in half. Well I'd say: OK no problem. You continue saying whereas '4' denotes no such division. Yes the correc...
No! Equality rules are spoken as Identity rules by mathematicians, it just happens that equality is used more: see this site on terminology: Glossary ...
I just wanted to add, that we can actually have a very simple system in which 2 + 2 = 4, that of first order logic and add to it primitives of identit...
Yes the reason is because I'm holding PA, and it shows you the rules about +, so I didn't want to go to all of that technical side. So I just mentione...
The two expressions are of course not identical, they are indeed distinct expressions, I already said that, that's clear because the expression 2+2 co...
Actually the somewhat acceptable definition would be as the Creator of the Universe. This assumes that existence of the universe begs a maker. This is...
There is no new law at all. It is a schema of statements, in first order logic it would be expressed as: that x = x, i.e. everything is equal (identic...
They are the same according to the game of identity called as "equality theory". There is a confusion here between expressions and what they denote, "...
With Actual infinity you have a set that contains EVERY natural number as an element, while with potential infinity you can only have finite sets of n...
there are no heresies here. Mathematics is not religion. And what you said was indeed harsh. It is not persons that we are discussing here, it is the ...
take any of lets say the meaningless rule following games. Add a rule that assigns to each symbol in that game a "particular meaning", this is a rule!...
Formalism is the point of view that mathematics is about Non-meaningful rule following games. More precisely put: mathematics is about string (of empt...
what I wanted to say is that "logical truth" is a relative concept, when you say that a statement s is logically true, I'll respond by saying "in rela...
I've already explicitly stated in my head post that I don't agree with formalism, mathematics is not limited to empty symbols rule following games, (u...
to me axioms are just a medium in which consequential truths are harvested. Yet definitely in mathematical practice the axiomatics develop by capturin...
analytic truth truth by virtue of the meaning of the words of a statement, synthetic needs meaning and correspondence with reality as well. Here with ...
it depends on what kind of arithmetic you have in mind. Anyhow you can of course list a complete set of axioms for arithmetic that is not recursively ...
I'll make a more elaborate response to this nice posting later. But for now I just want to resolve any misunderstanding that my prior response might h...
Thank you for your opinion about terminology about truth. But the essence of the matter of difference between us here (as it appears to me ) is just a...
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