How could the set of natural numbers not be finite?

[tautological]

The set of all possible streams of brain activity arising from all possible configurations of all possible neurons with all possible connections is finite, so if you accept that natural numbers are a creation of the human mind (brain), then don't you have to accept that the set of number is finite? When they start counting, mathematicians generally imagine the natural numbers as a platonic universe where objects are extended infinitely, but if thoughts about numbers are created by brain activity, and brain activity is finite, then every possible natural number and mathematical object imaginable from every direction with every one of thousands of subtle conscious factors is associated with a unique brain state, which is part of a larger finite set...

 

[Stephen Tashi]

The set of all possible streams of brain activity arising from all possible configurations of all possible neurons with all possible connections is finite...

There is no evidence for that since we don't know how the brain works. And neural activity consists of processes that take place in an apparent continuum of time and space, not in some finite discrete setting.

I think you should post threads that attempt to relate physics to logic in a different section of the forum because threads like this are usually locked or moved when they appear in the physics or math sections. Try the the PF Lounge.

 

[nomadreid]

tautological: Let us suppose, for the sake of argument, that the brain's capacity is finite. The question now touches on the Platonic/Formalist debate, which despite the death blow to Hilbert's Program dealt by Gödel, is not dead; only the classical Formalism is. Today's Formalism is concerned with how to divide mathematical structures into syntax and semantics. Taking a formalist stance, then, we can say that the words or symbols "infinite" and "finite" are either:
(a) syntactically, finite symbols for syntactical statements such as the symbol N in the axiom of infinity in ZF or its negation, respectively, or a statement such as "A set A such that there exists a bijection between the A and a proper subset of A", or
(b) semantically, the interpretation in a model of such symbols.
As long as there is no provable contradiction from these usages, then if they are useful, we keep them. From the viewpoint of a Platonist who does not believe that infinite quantities actually exist (whatever "exist" means), "infinite" may be thought of in the same way as school children are unfortunately often taught to think of complex numbers or infinitesimals: as something that is simply an imaginary go-between between a problem stated in real numbers and the solution stated in real numbers. There are also Platonists who believe that infinite quantities exist, who would then just tell you not to get the finger that points to the moon mixed up with the moon: the human brain could be finite, but it could point (via model theory) to something that was infinite. Finally, from the viewpoint of a staunch Formalist, even the interpretations are just symbols to be pushed around, and the symbols need not actually point at anything.
So, you pick your mixture of formalist and platonist (I don't think anyone is a pure formalist or a pure platonist any more), and go from there. A good starting point is to purge the question of the physics aspects which Stephen Tashi objected to by delving a bit into the theory of finite models and restating your question in those terms.

 

[ImaLooser]

The set of all possible streams of brain activity arising from all possible configurations of all possible neurons with all possible connections is finite, so if you accept that natural numbers are a creation of the human mind (brain), then don't you have to accept that the set of number is finite? When they start counting, mathematicians generally imagine the natural numbers as a platonic universe where objects are extended infinitely, but if thoughts about numbers are created by brain activity, and brain activity is finite, then every possible natural number and mathematical object imaginable from every direction with every one of thousands of subtle conscious factors is associated with a unique brain state, which is part of a larger finite set...

The human mind can't construct the set of natural numbers. But it can construct the concept of the set of natural numbers. The concept of the infinite is finite.

It is true that the set of thoughts ever thought is finite.

 

[nomadreid]

"The human mind can't construct the set of natural numbers. But it can construct the concept of the set of natural numbers. The concept of the infinite is finite. "

Well said, ImaLooser!

 

[micromass]

This does not meet the guidelines of the mathematics forum. The forum is for technical mathematics questions only.