Logic is how you are allowed to manipulate abstract symbols ; the rest of maths is deciding about some abstract symbol, and then playing with them a lot... Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
In my view N only has the meaning we give it through axioms ; axioms which are rules on how to write proofs.
What do you mean by 'meaning' ? :) Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
Axiomatization is not the prerequisite for mathematics, it's the endpoint. We have met the enemy, and he is us — Pogo
Also, is R more concrete than the set of p-adic numbers? is Euclidean geometry less abstract than other geometries ? Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
R is more concrete than the set of p-adic numbers. That is why it was invented centuries earlier.
And while Euclid and his contemporaries had axioms, mathematics had existed before them. The greeks may have invented the axiomatic-deductive method, but they did not invent mathematics. We have met the enemy, and he is us — Pogo
Which geometry is concrete to a blind person ?*
It seems you define concrete as intuitively accessible to the human brain... It makes god a very concrete concept nowadays. Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
It seems you define concrete as intuitively accessible to the human brain...
As for spherical geometry being invented after euclidean geometry, I don't know what came first, but spherical geometry was highly developed by babylonian astronomers while the Babylonian value for pi was still the integer 3. We have met the enemy, and he is us — Pogo
Also, is there mathematical truth independent of thought processes : is logic only a cognitive process ? Colman was contrasting logics with the rest of mathematics. Is logic "true" because it agrees with our thought processes - but many people think without adhering to the laws of logic. Why would logic be different from the rest of maths ? Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
Concepts of abstract and concrete can depend where you're looking at them from: N can be relatively concrete. In a moment we can consider what is concrete, precisely.
