I don't hold with the notion that mathematics can NEVER explain what happens within a singularity. It could simply be that we haven't evolved the full mathematical concepts to do so as yet.
Hawkings notion of imaginary time gives us a basis to form a 'no boundary' condition for time in higher dimensional reality, in the same way that the North and South poles represent no boundary edges to our concept of going North and going South.
Although some would say what's the use of imaginary time, there is a parallel in number theory where we make use of imaginary numbers. In my own field this enables us to do calculations with AC electrical theory that is much easier than using trigonometry. It also allows much more complex calculations to be done quite quickly and easily.
Another parallel is development of the Calculus, enabling us to work mathematically with values that change over time, such as velocity increase over time (acceleration). New mathematical tools allow us to work with problems in easier ways, or that weren't possible to work with prior to their advent.
It's not to say we WILL be able to find the complete math to deal with singularities but experience tells us not to be too quick to say something isn't possible. It was said that humans could never travel faster than a horse, then that we could never fly.
There may well be inherent limits to certain things. The speed of light appears to be one such limit within regular space-time. That's not to say anything is 'impossible'. It is predicted that light will travel slightly faster between a set of Casimir plates, due to the fact that the energy of the space-time between them has been decreased. This isn't a prediction that has been able to be tested by experiment, as yet. It has also been demonstrated that quantum tunnelling, across relatively large gaps, appears to happen instantaneously (or at least down to the limit of our ability to measure the time).
A singularity may well prove to be another limit, but there are indications that it is possible to develope the maths to maybe deal with that condition. In that respect, I believe we should be very wary of saying 'never'.
Hawkings notion of imaginary time gives us a basis to form a 'no boundary' condition for time in higher dimensional reality, in the same way that the North and South poles represent no boundary edges to our concept of going North and going South.
Although some would say what's the use of imaginary time, there is a parallel in number theory where we make use of imaginary numbers. In my own field this enables us to do calculations with AC electrical theory that is much easier than using trigonometry. It also allows much more complex calculations to be done quite quickly and easily.
Another parallel is development of the Calculus, enabling us to work mathematically with values that change over time, such as velocity increase over time (acceleration). New mathematical tools allow us to work with problems in easier ways, or that weren't possible to work with prior to their advent.
It's not to say we WILL be able to find the complete math to deal with singularities but experience tells us not to be too quick to say something isn't possible. It was said that humans could never travel faster than a horse, then that we could never fly.
There may well be inherent limits to certain things. The speed of light appears to be one such limit within regular space-time. That's not to say anything is 'impossible'. It is predicted that light will travel slightly faster between a set of Casimir plates, due to the fact that the energy of the space-time between them has been decreased. This isn't a prediction that has been able to be tested by experiment, as yet. It has also been demonstrated that quantum tunnelling, across relatively large gaps, appears to happen instantaneously (or at least down to the limit of our ability to measure the time).
A singularity may well prove to be another limit, but there are indications that it is possible to develope the maths to maybe deal with that condition. In that respect, I believe we should be very wary of saying 'never'.