Relax
Max is working himself to the bone. So hard that he is driving himself crazy and making himself sick. He believes that he has to continue to push through. The irony is that every time he takes the time to relax he finds clarity. This approach is what Sol has recommeded to the young man.
Garbage
After Max's computer is fried he takes the print out of numbers that it gave him and throws it in the garbage. He bellieves the numbers are worthless. The irony is that the number sequence the computer produced correctly prediccted the next day's stock market. Thus the machine he believes was broken works.
Coincidence
Max meets Lenny, a Jewish man interested in numbers at a coffee shop. He then sees him again, but he doesn't engage fully with the man. The irony is that Lenny is just as into numbers as Max is. Max learns of his interest in decoding the Torah and Lenny discovers that Max is a well known mathematician that can help him and he's been sitting right under his nose.
Finding an Answer
Throughout most of the film searching for a number, a sequence and he believes that can only be found with math, numbers. The irony is that he begins to find answers in nature as they relate to numbers such as in the shape of a sea shell which contains the golden ratio. Thus nature is what's producing this groundbreaking code not math.
Big Business
Max wants nothing to do with working on wall street. He believes it's only materialism. The irony is when wall street comes to Max with a microchip he takes it because it can produce the results he needs. He's now gone in bed with capitalism.
