~~~Actually, it is thanks to math that we know for a fact that the physical universe does not exist, at least not as we perceive it. "Physical" is what we can see, touch, smell, etc., but it's clear that those sensations do not exist and are only a creation of our minds.~~~
I'd say that the physical universe does exist if our senses exist. Our brain and senses exist, consciousness exists (well I do know my own consciousness exists, I absolutely cannot vouch for the rest of the human race), and a physical universe exists.
~~~it's clear that those sensations do not exist~~~
Please show me how those sensations do not exist. I sure believe they do, seeing as how I'm seeing, smelling and feeling at this very moment. I guess what I'm getting at is that I'm not too sure it would be an accurate statement to say: that which is constructed by our brain must not exist. Define exists.
~~~What "actually exists" is but a bunch of abstractions, some of which are expressed in mathematics.~~~
In this case the universe is your dream. You cannot prove that I myself am not just another abstraction. This is all in YOUR mind and I am not real. Well actually I'm "real" enough to you, just not a conscious entity experiencing my own life. You seem to be implying that there is no physical ... only mental. But if the universe is "inside", then so am I, and I exist only in your thoughts. However if the universe is "outside", then there is a physical that we all are experiencing.
Why would this have to be 'your dream' and not 'everybody's dream'? Occam's Razor (O.R. in this context - you'd be just as well to eliminate the more complicated unproveable possibility). You can never know with surety that there is another conscious mind in this universe other than your own any more than there being another universe with its own spacetime completely cut off, unreachable, nonintersecting, and noninteracting with our own ... or that a person in your dream at night is also consciously experiencing you. For all practical purposes scientifically and realistically, they are not real.
So either there's a physical universe, or you're alone right now...
I'll take the physical universe. (Hey, I KNOW I'm alive and existent, but then again, how do I know you're not in my mind? You may know you're not just as I know I'm not in yours, but if I were to be a figment of your abstraction, then my statement, "just as I know I'm not in yours," was simply you deceiving yourself.)