The math solution is, to find out properties of the things we work with, and prove those. Then we re-search those properties for more properties we can now prove. And on those more intricate properties, we build even more complex proofs.

In the case of your parking lot, the mathematician might start by asking: What do I know about this parking lot? The answer might be that it's the staging area of a factory where the finished cars are waiting to be shipped. The natural next question would be, whether the factory actually can produce orange cars. If we find that the answer is "no", we may continue to check other possible loopholes like the question whether other cars than freshly built cars from the factory are parked there.

The result is a proof along the lines of: This parking lot only contains cars of Ford model Ts, which always comes in black, so no car on the parking lot can be orange.

Ok, slightly contrived example, but you get the idea. For a look the other way round, take for example the proof that there are infinitely many prime numbers. It goes like this:

• I assume that there is only a finite number of primes.

• If that's true, I can multiply them all in finite time to get a product N.

• Consquently, N must be divisible by all primes.

• This means that N+1 is not divisible by any prime, when dividing I always get a remainder of exactly 1.

• Thus, N+1 is itself a prime, and my finite list of primes was not complete.

• This is bullshit. It contradicts my assumption. Since I didn't make a mistake in deriving this bullshit, my assumption must be bullshit. I conclude that there is an infinite number of primes.

You see, all this proof really does, is to derive other facts from the given facts. It did not need to look at every integer. It did not need to look at each prime. It just assumed that the opposite was true, derived some consequences (properties of the numbers N and N+1), and used that to show that the assumption was wrong. Much in the same way that above we didn't even look at a single car, we just checked some properties of the parking lot to determine that there is no orange car on that parking lot.

The trouble with this method is, that we can never prove everything that's true. That's another property of mathematical proofs that's been proven by Gödel. We can derive some astounding properties of many things we can formalize, but in the end almost all questions that are possible to ask require an infinite amount of time and space to prove them. Math is, by its very nature, restricted to those questions that have a finite proof which is actually easy enough for humans to find.

The math solution is, to find out properties of the things we work with, and prove those. Then we re-search those properties for more properties we can now prove. And on those more intricate properties, we build even more complex proofs.

In the case of your parking lot, the mathematician might start by asking: What do I know about this parking lot? The answer might be that it's the staging area of a factory where the finished cars are waiting to be shipped. The natural next question would be, whether the factory actually can produce orange cars. If we find that the answer is "no", we may continue to check other possible loopholes like the question whether other cars than freshly built cars from the factory are parked there.

The result is a proof along the lines of: This parking lot only contains cars of Ford model Ts, which always comes in black, so no car on the parking lot can be orange.

Ok, slightly contrived example, but you get the idea. For a look the other way round, take for example the proof that there are infinitely many prime numbers. It goes like this:

• I assume that there is only a finite number of primes.

• If that's true, I can multiply them all in finite time to get a product N.

• Consquently, N must be divisible by all primes.

• This means that N+1 is not divisible by any prime, I always get a remainder of exactly 1 when dividing by any number that divides N.

• Thus, N+1 is itself a prime, and my finite list of primes was not complete.

• This is bullshit. It contradicts my assumption. Since I didn't make a mistake in deriving this bullshit, my assumption must be bullshit. I conclude that there is an infinite number of primes.

You see, all this proof really does, is to derive other facts from the given facts. It did not need to look at every integer. It did not need to look at each prime. It just assumed that the opposite was true, derived some consequences (properties of the numbers N and N+1), and used that to show that the assumption was wrong. Much in the same way that above we didn't even look at a single car, we just checked some properties of the parking lot to determine that there is no orange car on that parking lot.

The trouble with this method is, that we can never prove everything that's true. That's another property of mathematical proofs that's been proven by Gödel. We can derive some astounding properties of many things we can formalize, but in the end almost all questions that are possible to ask require an infinite amount of time and space to prove them. Math is, by its very nature, restricted to those questions that have a finite proof which is actually easy enough for humans to find.