What's the meaning of "at most" in logic? I am having difficulty in finding meaning of the words "at most" in mathematics or more precisely in logic. It probably is to some extent a language and/or philosophy problem.
For example when we say "we can do ... with at most $x$ ... ". Does that mean that we can't do it with more than $x$ or does it mean that we can do it with less than or equal to x, but maybe also more?
 A: The word "with" is a bit ambiguous here. Suppose you said

We can nurdle any doohickey with at most $n$ widgets.

There could be (at least) two different, almost opposite, contexts.

*

*Widgets are something a doohickey has, which make it harder to nurdle. We are saying that provided a doohickey doesn't have too many widgets, we will be able to nurdle it.

*Widgets are used for nurdling things. We are saying we can always nurdle whatever doohickey we want, and we can guarantee not using too many widgets to do so.

In both cases we are not claiming the converse. It is possible in case 1 that we can also nurdle some doohickeys which have more than $n$ widgets, but it is also possible that we can't (or that we just don't know). In case 2, we can always avoid using more than $n$ widgets, if we want to, but perhaps we could choose to be wasteful and use more than $n$.
(You can make the statement unambiguous by replacing "with" with "having" for the first meaning, or "using" for the second.)
A: "At most" means that many or less.

I can fit at most 5 people in my car.
The auditorium seats at most 100 people.
The algorithm will converge in at most 1000 iterations.
The express checkout can be used with at most 10 items.

In all of these cases, the action can or will performed with X items or fewer. If you have more than X, the action is not possible (seating 150 people in the auditorium), or alternatively, the action will not occur with more than X (the algorithm converging in 1200 iterations).
