eli5: how do synthetic judgments a priori work in maths and physics?
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Since you can’t ‘expirience’ math all math except maybe the number 1 is a priori. But you can make judgments about mathematical things that are not already part of the terms you use. An analytical judgement would be that a triangle has three angles. What a synthetic judgment would be is a bit controversial in philosophy still but could be things like pi or a²+b²=c².
Are you asking about Kant’s views (as the phrase “synthetic a priori judgments” is strongly associated with him), or about modern views? I don’t know much about Kant, but nowadays there is a pretty big diversity of views in the philosophies of maths and science, which are usually regarded as being sharply distinct from each other. Physics is empirical, so the a priori stuff is all about how to conduct and interpret experiments and synthesise the information from them. In maths, you generally work within axiomatic systems (ones in which certain specific rules, called axioms, are assumed to be true) and deduce things ultimately from the axioms. Modern mathematicians are generally happy to jump from one axiomatic system to another and don’t necessarily believe that some are more valid than others. But obviously there has to be some reason for them to choose the ones they do, and there are various views about that.
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