We're struggling with a conceptual problem here. We don't need math for this one. The original poster is confusing the object he's looking at with the light rays bouncing off of (or emitting from) that object. I just went down this rabbit hole myself.
Our eyes have lenses and our retinas serve as the focal plane. No different from a camera or a telescope or whatever. Calling it a focal "point" is misleading insofar that our entire field of view resolves across the entire surface of our retina, not on one discreet point. A focal "point" should be thought of as a focal "plane" some specific distance from the lens. Not a single dot in space.
If you look at a tiny little point of light, perhaps a power button indicator on your PC or something. That tiny little point of light is emitting photons in all sorts of directions at once. The only photons from that tiny point of light your eye receives are the ones that happen to reach the surface of the lens of your eye. But the ENTIRE lens of your eye is receiving photons from that one tiny discreet point of light. If your eye didn't have a lens at all then that single point of light would resolve into a blanket of whitish opaqueness. (A "blur")
It's the lens of your eye that is bending all the light rays coming off that tiny crisp point of light that now blanket the entire surface of your eyeball lens back into that tiny crisp point of light representative of where the light originated from in the first place. It resolves back to that tiny point of light right at the surface of your retina where the information can then be sent along to your brain. And that single lens is performing this trick for your entire field of view and resolving your entire field of view into a beautiful crisp rendering over the entire surface of your retina.
Light scatters in all directions. A lens focuses scattered light rays back into something visually representative of their point of origin.
The lines you see entering lenses in diagrams don't typically represent the object, they represent the light rays coming FROM the object. Two different things.
Also, for a convex lens. If I have my eye at the focal point, would the image appear tiny? Since all the light rays converge there?
I think the error in this line of thinking should be apparent now. All the light rays coming into the optical system aren't converging to a point. They're focusing onto a plane. Only the light rays from a single tiny point of origin (like a power button or a star) are going to converge back to a tiny point. As for what you would actually see, it's important to remember that the optical system of your eye becomes an extension of the mechanical optical system you're looking through. The only focal plane that REALLY matters is the back of your retina.
Not to evade the question I guess. What would you see? I'm not entirely sure to be honest. I suspect you'd see a nice crisp rendering of whatever it is you're looking at. Much like looking at the surface of a flat screen television display.