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.

Izibalo ngamanye amaxesha zingabonakala njengamaphupha, ngakumbi ekuqaleni. Xasekukudala zide ziqheleke.  Ndakhumbula ukoyikeka kwalengcinga yeli qumrhu kum. Yaqala kancinci yakhula yade yakhula kakhulu ngoku zikhona nezinye. Kuwo onke lamaphupha  ngoku ndiyakhumbula , kukho elilodwa elinegumbi elikhulu eligcwele ngabantu, okanye umgodi, nam ndandisehla- okanye ndandisehla ngentambo?- Ndiziva ndifana nento engazange yabonwa aphemhlabeni. Iqumrhu apho into enje, eliqumrhu khalenze bavele ngaphantsi kunoba bebe lindelekile

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:

Ukongeza, amaphupha ne zibalo zinento ezininzi ezifanayo-Ndithetha amaphupha abantu, xa vukile, ukhumbule ubunalo ngoku ubulele. Nanga magama alandelayo

intangibility
brain-based ukhula kwengqondo
the telling is not the same as the thing itself ukuyithetha akufani noqobo lwayo
beauty/ugliness ubuhle/ububi
awe and fear ukuzithemba nokoyika
applied, but not directly causal yenziwe, kodwa hayi ngqo

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.

Kodwa ewe, izibalo ayilophupha. Singathi izibalo zine waka leminyaka zikhona. Umzekelo, we Pythagoren triples, unxantathu ukwakha  inani  le phawa ka (a+b). Ezinto zingabonakala njengamaphupha zivela kunye njengoba amaqumrhu evela kum, kodwa amaphupha ngawe thutyana, ezi zihlala ixesha elide njengayo nayiphi into eyenzwe ngumntu.

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?

kodwa lento yohlala ixesha elide kwezibalo, izibalo zifana zodwa, uyenze ibeyinyani nje ngamaphupha , okanye ibeyi nyani kakhulu? okanye omnye angathi amaphupha achaza ubuyena nomphuphi, kodwa izibalo azinabunyani ngaphuzu kwamaphupha?  

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).

Amaphupha ayinyani? avakala eyinyani, buninzi ubugqina obusuka EEG ebubonisa ukuba into eyenzeka engqondweni emntwini, ekuhambeni, bathi icitha ukuphupha. kodwa akhoonto yocitshilela amaphupha xa esenzeka. Yonke into ethethwayo ngephupha yinkumbulo. Iinkumbulo aziyonyani konke konke. Bathi inkumbulo iyafana ihlala ingeyo nyani: sonke sakhe samva umntu ethetha ibali kabini lingafani. Iveziwe lento kwizifundo ezininzi zepsychology: aythembakali kakhulu( ayithethi uba ayinamsebenzi, ingcinga angathembekiyo ibhetele kune ngekhoyo).

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.

Inyaniso ngamaphupha ayingowo thenjwa, uba sifuna ukuthi inkumbulo ngqo yinyaniso. Soze ukwazi uthetha iphupha ngohlobo elilo ngqo.

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.

Inyani yamaphupha isemaphupheni ngqo: ikhona into ekhoyo, iyinyani ngalo mzuzu. Inkumbulo ayikho ncam nje ngoqobo lwalo. Obubungqina bungacaciyo buyabonisa nyan ikhona into ibikhona, noba umntu akanoqiniseka ukuba ibiyintoni na.

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.

Ubunyani makube sekubeni kwenzeka ntoni ngoku  ngalo mzuzu kuzo, ukubhala phantsi bobona bungqina. Le yinyani nje ngobala apho umntu avakakuhle ubude nodibanisa ubungqina  emva kwe yure, kwentsuku, iiveki zofunda. Uba izibalo aziyonyani nakwelonqanaba , zange zibeyinyani. Nje ngephupha, ezozinto zingakhawuleza zicime. Kodwa hayi nje ngamaphupha, zinga phinda phinda, ndingaphinda phinda oko koko, kwenzeke into eyiyo. Okanye, iphantse ibeyiyo.

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?

Izibalo ziyinyani kwiqondo lobunyani? ukubulawa kuka Polonius ngu Hamlet buyinyani qho xa buvela, suyihoya ngabadlali aba akekho owabulala nowa bulawayo.

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.

Amaphupha kawizinga le Frikhishini? Kungathwa avela ngaphandle kwamandla kumphuphi, hayi nje nge mpinda mpinda yama bali.  Kwenza umdla uba ababhali abaninzi bathi abadlali bafika begqityiwe, futhi bazisa  msinya uba isitori esitsha singayinto yexeshana okanye yexesha elide. Basose becacisa into ezintsha engqondweni yabo zibufana nemidlalo. hayi zonke. Ibali into ephuma emva kwexesha elide lobhala. Ezinto zintsha ekugqibeleni zifana kakhulu namaphupha, ukubhala kungafana neengcinga zamaphupha. Okanye ubuchwephetsha bobhala benza kubelula kakhulu futhi bucacisa amabali ngaphezu kobalisa iphupha ekuseni.

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.

Amaphupha awayiyo ifrikhishini, nangona uchaza kwawo ungayiyo. Amaphupha ayinyani ngoku gqibeleleyo kumphuphi xa ephupha, kodwa  obobunyani buphela xa bevuka- okanye, singathi umntu ukwimo apho engenokwazi uchaza umehluko phathi kwephupha nobunyani.

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.

Izibalo, kwelinyi cala, singathi zithembakele kwingqondo echubekileyo. Sikwi qondo eliphezulu, apho xa sibalisa ibali lezibalo livakale, libali elinye maxesha onke. Okanye, libebufana kakhulu, umehluko ukhawuleze ubonise uba ibali alivakelanga.

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.

Obo, bubunyanni bezibalo obulindelekileyo esibakha ezingqondweni zethu( nawo apmaphupha) ngalonto zingakwazi ukwakheka, ngoku gqibeleleyo, futhi ihlale amawaka eminyaka, naku tshintsho olukhulu kulwimi, nakwi hlobo lobonisa amasiko. Nomculo soze uyifune lonto.