# What arguments are used by Rationalism to establish the subject and predicate terms of mathematical propositions possess different meanings (i.e., that these propositions are synthetic), and that the truth (or falsity) of these propositions is known a priori?. Philosophy

When we analyze the subject we are given each part separately. We have to leave off analysis and consider the nature of each number in the equation and then perform the intellectual act of combining the numbers to form the correct answer. Rationalists want to prove that mathematical statements are synthetic statements known a priori and that the subject and predicate terms have different meanings. For example, take a complex problem such as 2,789 * 127 = ?. One can understand the first meaning to the left of the =, but not necessarily what the answer is, the second meaning. Therefore, one can know one meaning but not the other/understand the symbol of one without the other, proving there are two meanings, making it synthetic. Mathematical statements always achieve certainty and that the answer will never change. Therefore, mathematical statements are a priori.

Knowledge a posteriori never gets you to certainty, it always falls short of certainty.

They believe that anything conceivable is possible.

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