Plato was a superb mathematician. In mathematics (especially in geometry) you can imagine abstract shapes with extremely well defined properties. One amazing thing about mathematics is that you can approach the subject from a large number of different perspectives and come back to many of the same forms. So, if you started math by studying numbers, you would discover many of the same things that you would have discovered if you started with geometry.
Needless to say, mathematics leads itself easily to logical analysis as Euclid would later demonstrate.
It is possible that Plato thought he would be able to find the same type of well defined forms in matters of ethics and political theory. His version of the Socratic dialogues seemed to be aiming at finding universal definitions of ethical issues.
The problem, of course, is that, when people engage in Socratic like dialogues, they can come up with all sorts of bizarre definitions for ethical terms. For example, a state of peace would happen after I kill all of my enemies. Therefore, my killing my enemies is an act of peace. (Many of the modern groups claiming Peace and Social Justice have very bizarre definitions of peace and social justice).
Here is what I see: When you do free form open enquiry about mathematics, you keep coming back to the same forms. When you do open enquiry about matters of ethics, you end up with a bunch of disparate and conflicting definitions.
From this observation, I would say that people can be loosy goosy in the logic they use when studying math, but that people need to be more disciplined in their study of ethics, politics and legal matters. For diplomacy to work, there must be a concerted effort to make sure that people are using the same definitions.
When mathematicians use different definitions, it generally becomes obvious because people keep coming back to the same basic Platonic forms.
Mathematics is bizarre. It is a subject that benefits when people engage in totally free form open enquiry. It is a subject that benefits when people run off into a corner and create a new language to describe the subject. What happens is that this open enquiry comes back to the same basic forms. The people creating a new language for math sometimes uncover things that the last language missed.
The same free form open enquiry does not work in politics. When the political class use different definitions, they end up creating a warring divisive society.
Politics, ethics and many areas of physical science require much more discipline than mathematics.
What is interesting about the modern era is that the intellectuals came up with needlessly terse method of talking about mathematics called symbolic logic, and they ripped the study of informal logic from the classroom.