This comes from the question "Is human intuition consistent with current structure of mathematics?" I asked myself.
Intuitively we agree on very many things and it seems we can "prove" almost everything. My example of that:
When someone says "It is cloudy" and you see clouds you agree with that. In fact you might even think the world is an illusion and clouds, your eyes, light emitted to your eyes, your eyes themselves, your brain and brain signals are illusions and so on. But you still agree that it's cloudy. So why?
In very deed I reject the notions of good, bad, right, wrong, true and false themselves. What I only accept is the sense. Different propositions, ideas, forms and actions might make more or less sense. Thus, when we agree on something we just mean what we think and say makes sense.
I think it is utterly difficult to define our intuition formally. It makes sense to say that our intuition is too complex to be described just by symbols. This led me to the thought that sensualistic logic is far more expressive than anything other we could think of.
Example of sense in mathematical meaning:
There were times when equation x^2=-1 was thought not to have solutions. So, no solutions exist meant "true". But now we say it is false. How can it be both true and false. The thing is that both "do not exist" and "exist" can be true. If you want to escape such ambiguity, you probably need to ask infinitely long question. But whrn we say about sense, we don't need to do it. Maybe we somehow can understand the context (which must be infinitely long to escape ambiguity) and give an answer according to it.
But mathematics are built on the notions of truth and false. Eventually that resulted in Gödel's theorems, Post's theorem an so on. These theorems state that nothing can be proved to be true. And still we agree on things, we think some of them make sense.
Another issue with mathematical foundations is that we try to define them formally. And my logic rejects the concept of formality. Is that an issue for mathematics?
It seems all this mean mathematics in its current form can't describe our way of thoughts. And assuming materialism that would be equal to say that mathematics in it's current form can't describe reality and define Theory of Everything. And if it's so, we need to change foundations of mathematics to advance further in physics.
But I'm wondering if my solution itself negates mathematics in it's current form, because without defining the truth we can't prove Gödel's theorems as they would involve undefined concepts. Actually, I even thought about self-referential propositions in my logic.
The proposition "This proposition does not make sense" indeed does not and the proposition "This proposition makes sense" does. So, self-referential paradoxes do not exist in sensualism.
But how deep are the differences between my logic and current logic of mathematics? Could we even define arithmetics using such logic? More interestingly, assuming my logic, would all the Millennium Problems retain their sense? And if no, then do Millennium Problems have sense right now?
What would be the consequences for science under this logic? Could that help us to advance further in science?
Logic described here must not be confused with intuitionistic logic or paraconsistent logic.
Examples of my logic:
A makes sense. Not A makes sense. Therefore, A and not A makes sense. A or not A makes sense as well.
A does not make sense. Not A does not make sense. Therefore, A or not A does not make sense. A and not A does not make sense as well.
A makes sense. Not A does not make sense. Therefore, A and not A does not make sense. Contrariwise A or not A makes sense.
In the case of several variables these rules might not work. Consider "Sun is large or bright". Actually, it does not make sense because it tries to find connection between the two, but there is none. But if we put 'and' instead of 'or', the proposition becomes meaningful, as 'and' may be used to link different properties.
Thus I assume such logic rejects negation in it's standard form, but does not reject the idea of negation.