One of the obvious ways computers are limited is in their representation of numbers. Since computers represent numbers as bit strings of finite length, they can only represent finitely many, and to a finite degree of precision.
Is it a mistake to think the humans, unlike computers, can represent infinitely many numbers with arbitrary precision? We obviously talk about things like the set of all real numbers; and we make use of symbols, like the letter pi, which purport to represent certain irrational numbers exactly. But then I'm not sure whether things like this really do show that we can represent numbers in a way that is fundamentally beyond computers.
Imagine that a Greek philosopher promised to his queen that he would determine the greatest prime number. He failed. Do you think that the mathematical fact that primes are infinite was a cause of his failure? I'm asking this because I guess most philosophers think that mathematical facts have no causal effects.
My understanding is that we can use systems like Peano Arithmetic to prove the seemingly basic truth that 1+1=2. Do such proofs actually give us reasons to believe that 1+1=2 that we didn't have before? Are they more fundamental or compelling than whatever justification a mathematically-naive person would have to believe that 1+1=2?
I am interested in how mathematical propositions relate to objects in the world; that is, how math and its concepts somehow correspond to the physical world. I have thought a bit about the issue, and realize that what happens, say, with numbers when we do some kind of mathematical operation with them may be the same as when we deduce one proposition in logic from another (If there is a number 2 and an operation "+", and an operation "=", then the result of using 2 + 2 = 4); but my question is this: does the proposition 2 + 2 = 4 mean the same thing as taking two objects and placing two more objects alongside of them, and then counting that there are four objects?
I have a question. Years ago me and two friends got into a debate about a riddle. The riddle goes like this:
A train starts from point A and is travelling towards point B. A wasp is travelling in the opposite direction at twice the speed of the train, the wasp touches the tip of the train and goes back to point B. How many times does the wasp touch the train?
(this may be one version of many, but this is how it was told that faithful evening)
So the "correct" answer was, infinte times. (similar to Zeno's paradox with Achilles and the tortoise).
I said, well in theory it's infinte times, but if you were to actually do it, the train would hit point B eventually so it can't be infinte times? For it to be infinite times it would have to stop time (or something)
So what would happen if you actually tried this? Say we do an experiment with a model train and instead of a wasp we use a laser (for accuracy). First we measure the railway track and only run the train, let's say it takes 10 seconds to go from...