I am not referring to two people or two cats or two pebbles or two cells etc.... That is two as an adjective. It is describing something concrete. I am talking about the number "2", the noun. I am talking about the number 2 as an abstract concept that exists purely as an idea. When a person counts: 1, 2, 3, 4, 5,... they don't have to be counting objects. They can just be counting. This is what I mean by the number 2. It's the idea that numbers exist separately from the natural world purely as concepts. There is a level of abstraction going from two people to simply 2 (actually there are two levels of abstraction ).
Traditional thinking is that abstract thought is what separates man from beast. Perhaps apes or dolphins can conceive of the number 2 as a separate abstract entity? That I don't know. I very much doubt a dog or cat can. Any understanding that a dog or cat has of the number 2 is concrete and not abstract.What is sight, in essence? Sight is cognition.
Cognition is the basics and the origin of biological life. Later, cognition created the brain, the organ of sight.
The word sight means literally cognition. Do you SEE what I say? To see is to understand, to know.
The brain did not create cognition. What did the first cell do? It di-vided. Di < Dis > apart > a part: originating in the concept 2.
.. videre < uidere was a lost verb already at the time of the Roman Republic. The origin is the Indo-European root WID = to know > to see.
The cell does not know what it is doing when it divides? Our cognition is the cognition? Our mathematics is the mathematics? Only Homo Sapiens Sapiens can understand the concept of 2?
Dogs and cats believe their master and mistress constitute one entity?
The concrete precedes the abstract, but the concrete is not mathematics. In the most rigorous sense mathematics is logical proofs. Whenever a math major takes their first "proof" class the professor often says, "Welcome to your first actual math class." This is why I say pure mathematics is mostly art, because constructing mathematical proofs is more of an art than a science.Intelligence suddenly erupted in the human brain?
An architect or a carpenter do not need to be concerned about the curvature of the earth. It has no relevance in their little world.
The Sumerians found mathematics as a by-product of astronomy 1800 years prior to the Greeks. They invented the sun clock. They did not assume the world was flat.
Does the concrete precede the abstract?
Yeah I think wildcat is coming at it from a more historical perspective and I am looking at it as more of a logical framework. I can see it both ways, but the thing is that over time mathematicians have gradually put more emphasis on rigor. Eventually they desided to start over from the very logical foundations and prove everything from there. The history of mathematics is more of a novelty to mathematicians. Knowing the axioms, postulates, defintions, ect... and how to derive the various theorems from them is what is important.Originally Posted by Xander
My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14 - August 14)
http://www.revoltingvegetables.com
You are right.
We perfectly agree.
You said it, again.
Your mathematics is but art.
Is linguistics an art?
No. Why?
Art is a free sport.
Linguistics is about what is, in the line of semantics that is, and therefore it cannot be free.
Linguistics is subject to the basic laws of mathematics.
In essence, linguistics is only a branch of biology.
Biology is about mathematics.
Not our mathematics, though.
We agree in our disagreement. Our mathematics can never be mathematics = science. It is art, as you say.
Irony is the paradox, or it looks like it.
A Play in One Act
WILDCAT (in the tent of the fish monger woman, near the harbour):
- Two fish, please.
THE FISH MONGER WOMAN:
- What! How dare you insult me! Am I a beast now!
Two has nothing to do with fish! It is an abstract number!