[reposted, with additions, from 22 November 2012]
The film suggests that Lincoln was using mathematical concept of transitivity as a guiding principle of political consensus-building.
A more detailed discussion of political aspects can be found in Ravi Chaudhary‘s post on Huffington Post blog, and a remarkably interesting criticism (by Christopher S. Morrissey) in The Catholic World Report:
[…] screenwriter Tony Kushner portrays Lincoln’s pursuit of the Thirteenth Amendment as flowing, not from Christian charity, but from mathematical reasoning analogous to the abstractions Lincoln read about in Euclid’s Elements.
“Euclid’s first common notion is this,” says Lincoln in the film, “Things which are equal to the same thing are equal to each other. That’s a rule of mathematical reasoning. It’s true because it works. Has done and always will do. In his book, Euclid says this is ‘self-evident.’ You see, there it is, even in that 2,000-year-old book of mechanical law. It is a self-evident truth that things which are equal to the same thing are equal to each other.”
The scene is a fiction. The truth is more interesting. Lincoln himself actually said this: “One would start with confidence that he could convince any sane child that the simpler propositions of Euclid are true; but, nevertheless, he would fail, utterly, with one who should deny the definitions and axioms. The principles of Jefferson are the definitions and axioms of free society. And yet they are denied, and evaded, with no small show of success. One dashingly calls them ‘glittering generalities’; another bluntly calls them ‘self-evident lies’; and still others insidiously argue that they apply only ‘to superior races.’” (6)
Difficult as it is to teach someone mathematics (and to apply its self-evident truths in a process of reasoning), it is even more difficult to teach and apply the truth of the Declaration of Independence about human equality (“that all men are created equal”).