Why can't our cognitive sense or five senses sense hyper-dimensional spaces?
Is it because of the way our human body is built?
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Because we have not evolved such abilities, because they would not be of (evolutionary fitness increasing) use to creatures like us. This is similar to how we lack senses for radioactivity or gravitational waves.
Our space-time is to a good approximation 4+1 dimensional (three space dimensions and one time) even if some superstring or brane theory were to hold true and there were extra dimensions. But since these are "rolled up" to a tiny size they do not provide extra information or degrees of freedom we can use.
Organisms in a universe with more big dimensions would presumably gain survival advantages from perceiving and acting on them... assuming there was something there to perceive and act on. Earth-life isn't strongly affected by gravitational waves (no nearby strong sources) and cannot affect them (we are too low-density and slow), so there is no pressure to evolve these abilities. Similarly there has to be some advantage for the high-dimensional creatures to exploit the extra dimensions for their evolution to give them such abilities (if they are physically possible).
It is a bit less clear why we are bad at thinking about high-dimensional spaces. That our brains are wired to be good enough at interpreting sensory information from our world is obvious, but we humans do have cognitive abilities that allow us to visualise and think about non-physical concepts such as causal links, mathematics or abstract spaces. A likely answer is that the general intelligence we have allows us to do things that are not evolutionary pre-specified and this includes abilities that do not directly boost survival: much of our thinking and acting is about concepts distant in time, space or possibility (and done well will increase individual evolutionary fitness). These general abilities allow us some abstraction ability but is not tuned to make it perfect. Hence we are better at some abstract mental tasks than others, and high-dimensional thinking is one of the tasks we are fairly limited at.
Senses and brains reflect the environment they evolve in, but do not have to make use of all available information or describe the environment perfectly - they only need to be adequate enough.