You can write 'cannot', can't you?

With can things are much easier than with will. No language history, no exceptions,...

I remember my written exam in mathematics in year 12 of my modest school career. I was sitting there under shock because the exam tasks I had just read had given me a rough idea of how many lessons I had spent daydreaming. With some effort I might have caught up with the others in five months, but all I had now were five hours.

In this situation the exam supervisor - a teacher of German - asked us
Can I help you?
and in my frustration I blurted out
I doubt that you can, but you may.
He looked at me in a manner that did not exactly express paternal love and I suddenly realized that mathematics might not be my only problematic exam.

The common misunderstanding I told about earlier is reflected in this little story. It is the question, which modal concept can expresses.

Originally there was only the concept of ability. This can be found in many everyday sentences like

can modals

As long as the concept of ability is to be expressed, the analogue for can is [be] able to. The square brackets around be indicate that this word has to be changed to reflect tense, person and number.
When I was younger I was able to ... (enter your fondest memory here).

However, very often we use can with the concept of permission, like in

The analogue forms for the concept of permission are [be] allowed to and [be] permitted to. This is why I prefer languages to natural sciences because here language shows its flexibility and that it can develop and change if necessary. Or, in simpler words: If enough people make the same mistake long enough, they will be right some day. Try this with mathematics!

Before we forget:

It can also happen that can expresses a possibility. If you miss an example, read these two sentences again because it can happen that the examples are integrated in the text. Got it?

However, things get difficult whenever we need an analogue for the concept of possibility. The sad truth is: There is none. So if you need to express
There can/may/might be a problem.
in past tense, your only solution is to use could/may/might combined with a present perfect.
There could/may/might have been a problem. (Technically, the sentence There can have been a problem. is also legal, but it sounds unusual.)

This seems to contradict everything we learned in the tenses section, but fits to the rules for conditional complex sentences perfectly well.

The final words

in this section address the word could. This word may be the past tense of can or its conditional form. The remarks under would (click here) apply to could as well.

And from here you can go back to the modals table.