This is not as simple as it seems. Please bear with me.
I'd say that the physical universe does exist if our senses exist.
It depends on what you mean by 'our senses'. If you mean things like 'vision', 'hearing', 'touch', then you are right, it's silly to argue that those things don't exist. But that's not what I said anyway, all I said was that those things are not "physical".
On the other hand, if by senses you mean "eyes", "ears", "brain", then you are in the extremely unjustified position of asserting that your senses exist because you can sense them. That's a circular argument.
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.
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.
I would define "existence" as a property of experience which is unrelated to my conscious will. If I start singing in my head, the music is not real; I can start and stop it, change the pitch, the lyrics, do whatever I want. I know something does not "exist" when I perceive a strong correlation between my perception and my will. On the other hand, I can't do anything about the muzak in my favorite restaurant; my experience of the music does not correlate to my will. The muzak in the restaurant exists, the song in my head doesn't.
So far so good, I suppose. But it can be shown that the act of perception is essentially mental in character. Muzak doesn't really exist, only sound waves do. It takes a mind to convert sound waves to music. However, by the same reasoning sound waves do not exist, only changes in air pressure... only air... only atoms... only quantum particles... only what?
The world as perceived by you is a mental construct, but the world as it really is is not a mental construct. So what is it then? The simple answer is, things you cannot mentally construct.
In this case the universe is your dream.
Nope. Dreams are also mental constructs. The real universe, as it really is, cannot be seen, heard, touched, it can only be understood in abstract terms, such as the ones used in physics.
You cannot prove that I myself am not just another abstraction.
You are as much an abstraction as I am. That is what makes you real (remember, reality is abstract)
This is all in YOUR mind and I am not real.
Actually, the abstract ideas in my mind are absolutely real. It's the things "out in the world" that are not real.
You seem to be implying that there is no physical ... only mental.
That's not quite correct. The words and pictures you are currently seeing in your computer screen are "physical", but they are not a correct description of what is really going on. I suppose you know the story about electron beams and cathode-ray tubes. I suppose you also understand there must be yet another story behind that one. It's stories all the way to the end. The stories are real, the images and sounds that help you understand them are not.
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.
If the contents of my experiences are private to me, if you can't see what I see and hear what I hear, how is it that you know we all experience reality the same way? What if my experience of reality is so foreign to you that you can't even conceive of it?
Here's the thing. You think we all share the same reality for a very particular reason: because the abstract concepts we communicate, the stories we tell each other, make some sense. And the reason the stories make sense is because we all share them. We don't share our senses, we only share the stories they tell us. Reality is the story our senses tell us.
So either there's a physical universe, or you're alone right now...
Actually, it's quite the opposite. Either our minds are connected through abstract ideas, such as love, friendship, understanding, rather than sensations, such as brown eyes, white skin, low voice, or we're all completely separated from each other. If you ever felt lonely in the middle of a crowd, or if you ever experienced a deep connection with a person without saying a single word, you know what I'm talking about.
Don't be bothered by the fact that your perception of the physical world is just an illusion. The world of the abstract, of the mind, is so much richer.