Throughout their history, arithmetic and mathematics in general have largely been held as peculiarly separate from the physical and social sciences. This is based on the seeming certainty of arithmetic, opposed to the verifiability of the empirical sciences. This uniqueness has generated significant discussion on the exact nature of mathematical truth. The inductivists held that mathematical truths were simply an extension of empirical science; mathematical truths are simply highly probable rather than certain. Hempel defends a view known as logicism: mathematical truths are analytic truths of logic. This paper will analyze both viewpoints in order to demonstrate logicism's superiority.
The paper will begin with dismissing the notion …show more content…
This inductivism holds that mathematics differs from empirical sciences in two ways; mathematics are more general than empirical sciences and also have been tested and confirmed to a greater extent than the other sciences (Hempel 543). According to this view, mathematical theorems are not significantly from scientific theorems in general; mathematical theorems are generalizations of past experience (Hempel 544). This means that mathematical theorems are empirically verifiable; if we ever found a scenario that 2+2=5 than we could possibly abandon the theorem that 2+2=4. Mill makes the case that mathematical truths are simply extensions of the results of empirical …show more content…
Although logic and mathematics are analytic in nature, they are valuable conceptual tools (Hempel 553). Mathematics and logic function as a form of theoretical juice extractor; they can produce no more factual information than that found in the assumptions but they produce more information than initially expected (Hempel 554). Within an experiment, the answer is always theoretically contained before it is opened up to mathematical and logical analysis; the answer that these tools provide are psychologically useful for us (Hempel 553). Mathematics lack empirical content and yet are a necessary and effective tool for the understanding and mastery over nature (Hempel 555). Thus, Hempel is perfectly capable of how analytic mathematics can work with synthetic empirical
The paper will begin with dismissing the notion …show more content…
This inductivism holds that mathematics differs from empirical sciences in two ways; mathematics are more general than empirical sciences and also have been tested and confirmed to a greater extent than the other sciences (Hempel 543). According to this view, mathematical theorems are not significantly from scientific theorems in general; mathematical theorems are generalizations of past experience (Hempel 544). This means that mathematical theorems are empirically verifiable; if we ever found a scenario that 2+2=5 than we could possibly abandon the theorem that 2+2=4. Mill makes the case that mathematical truths are simply extensions of the results of empirical …show more content…
Although logic and mathematics are analytic in nature, they are valuable conceptual tools (Hempel 553). Mathematics and logic function as a form of theoretical juice extractor; they can produce no more factual information than that found in the assumptions but they produce more information than initially expected (Hempel 554). Within an experiment, the answer is always theoretically contained before it is opened up to mathematical and logical analysis; the answer that these tools provide are psychologically useful for us (Hempel 553). Mathematics lack empirical content and yet are a necessary and effective tool for the understanding and mastery over nature (Hempel 555). Thus, Hempel is perfectly capable of how analytic mathematics can work with synthetic empirical