Mathematics can sometimes seem dream-like, at least on first encounter. Later on, one gets
used to a new mathematical object, and it seems everyday. I remember how strange the idea of
a group was to me, how mysteriously it grew from three almost trivial axioms to a forest
of subgroups and quotient groups and equivalence classes and so on. Of the few dreams I
now recall, there was one with a huge hall full of people, perhaps a giant cave, and I was descending a long,
rickety staircase — or was I sliding down a cable? — feeling myself among a heretofore
completely unsuspected part of humanity, who perhaps nobody from above ground had ever seen.
Groups were a bit like that, and saying that a square had a symmetery group did not
make them appear any less unexpected.

Furthermore, dreams and mathematics have a lot in common—I mean here the dreams that
people, when awake, remember having had when asleep. Here’s a partial list:

intangibility
brain-based
the telling is not the same as the thing itself
beauty/ugliness
awe and fear
applied, but not directly causal

But of course, mathematics is not a dream. In particular, we can point to mathematical objects
that have existed for thousands of years. For example, the lists of Pythagorean triples from
Babylonian clay tablets, the proofs that now bear the name of Euclid, the triangle for constructing
coefficients of powers of (a+b). These may well have appeared as
dreamlike to their contemporaries as groups appeared to me, but whereas dreams are famously ephemeral, these
are as enduring as anything humans have made.

But does this enduring quality of mathematics, which is surely based on the fact that
mathematics is uniquely communicable, make it as real as dreams, or more real? Or indeed
might one want to say that dreams always to some extent reflect the reality of the
dreamer, whereas mathematics is so completely abstract that it must be less real
than dreams?

Well, are dreams real? They certainly feel real, and there is lots of evidence from EEG traces
that something happens in the brain during times that people, upon waking, say they spent
dreaming. But unfortunately there is currently no way of recording dreams while they occur. Everything
one says about a dream is some sort of memory. And even more unfortunately, memories are not very real
at all. The subjective sense that a memory when recalled is always the same is just not true:
we all have experience of hearing some-one tell a story twice with different `facts’. This
has been demonstrated as a general feature of memory in many psychological studies: it is
very unreliable indeed (which is not to say that it is useless, even an unreliable memory is
better than none).

So the reality of dreams are dubious, if we want to base that reality on accurate memory. There
is absolutely no way of telling whether the dream content that is remembered upon waking is the
same as during the dream itself.

The reality of dreams must be based on the dream itself: something is there, it is real at that
moment. The recall is a pale echo of that reality. This feeble trace is nevertheless evidence that
something really did exist back then, even if one cannot be quite sure what it was.

Likewise, the reality of mathematics must be based on what happens at the moment of doing
mathematics, of which the writing down is mere evidence.
This is as true for simple acts of counting as it is for those all too brief
moments when one understands a long and complex proof after hours, days, weeks of intense study.
If mathematics is not real at that point, it is never real at all. Like a dream, those things may
quickly fade. But unlike a dream, they can be repeated, indeed they can be repeated over
and over again, with a claim to perfect fidelity of reproduction. Or at least, as close
to perfect reproduction as is humanly possible.

So is mathematics real in the sense of stage reality? Hamlet killing Polonius is real every
time it is enacted, never mind that the actors themselves are neither murderer nor victim. Is
mathematics just another variety of fiction?

Well, are dreams a variety of fiction? It could be said that they arise without any volition
in the dreamer, unlike the deliberate construction of stories. It is interesting that many
authors report that characters arrive apparently complete, and that they perceive very quickly
whether a new idea for a story should be worked up into something short or long. Nevertheless,
they also clearly do not claim that these initial stirrings in their minds are equivalent to stories.
Not at all. The story is what comes out of long process of writing. Indeed,
these initial glimpses of the final fiction may have a lot in common with dreams, and the writing
may be similar to the construction of the memories of dreams. Perhaps the technology of
writing makes possible far longer and elaborate stories than can be told the morning after a dream.

So dreams are not fiction, though perhaps the reporting of dreams may be.
The dreams themselves are real enough to the dreamer at the time of dreaming, but that reality
is gone by the time they wake — if not, we would say that a person is in the unfortunate
state that they cannot tell the difference between dreams and reality.

Mathematics, on the other hand, can be reliably recreated in a trained brain. We are in the
extraordinary position that, when we tell a mathematical story in such a way that it
is fully understood, it is the same story each time. Or at least, as much the same story as
is humanly possible, in the sense that any difference is immediately evidence that the
story has not been understood.

That is, mathematical reality is exceptional among all the realities we create in our brains (including
dreams) in that it can reliably be reproduced, with complete accuracy, and endure for thousands of years, across massive
changes in language, culture and modes of expression. Not even music can claim that.