Induction, as a method of generalization of knowledge, is important concept for both philosophy and mathematics. Sometimes it serves the same way for philosophy and math, sometimes it does not. Here, I am going to talk about how inductive reasoning serves for mathematics and when it does not.
Mathematical induction is really nothing like empirical induction, but is instead a deductive technique[1]; it is unambiguously a form of deduction. Mathematical induction, with a common example, always says that if a generalization is true for k point, then it must be true for k+1 point. However, what I suggest is more simple; everyone may guess that math must obviously be inductive, because when we are doing math, what we want to do is to find out more general expressions, which is infinite, through more specific cases, which is only one. For example, “every even number can be divided by 2” gives an impression of how come we are sure that a very big even number will not break this rule? This comes from the logic of mathematical induction. Because, if we prove that the smallest even number is 2 and we comprehend that every even number after 2 is also divisible by 2 –even we did not examine this rule for every even number after 2 –actually we do not have to. This is how our knowledge of an infinite set of unexamined cases can be as certain as the conclusion of a valid deduction, quite unlike the conclusion of an ordinary induction. Thus, mathematical induction, one mathematical rule, works on ad infinitum. Then, mathematics itself is giving the idea of inductive reasoning in point of its working style.
As for why mathematics suggest more inductive instead of deductive is a bit about philosophy history[2]. Francis Bacon, as the establisher of inductive knowledge method, abnegates deductive reasoning because it’s being absolute cogitation method. On the contrary, Hegel, as a pure idealist, supports deductive reasoning, because he claims that the only way for idealism is deductive reasoning. As we can see from this historical conflict, inductive reasoning came into more prominence in mathematics; mathematics has nothing to do with idealist way on the road.
No wonder mathematics is lawlike, especially when it is on the way of induction; but sometimes it does not work. For example, it does not give any idea about Riemann Hypothesis[3] is true. So, for this kind of advanced mathematics, mathematics has to be created a new concept to catch some evidence about its correctness. This new concept is called non-deductive logic, which does not really stand for inductive logic –but still have some taste inside. This basically stand for logical probability as James Franklin mentioned in late 80’s[4]. Actually, he says that “Occurrence of non-deductive login in math is a big embarrassment”.
Various inductive principles obviously give results and are not simply dismissed as pragmatic or heuristic. Yet we can suppose that they are not principles of logic, but they work because of natural laws. However, this is not available in mathematical case. Because we accept that mathematics is true in all worlds; any principles guiding the relation between hypothesis and evidence in mathematics can only be logical. Thus consideration of a mathematical example reveals what can be lost sight of in the search for laws: laws or no laws, non-deductive logic is still needed to make inductive inferences.
So far, as from both cases, inductive reasoning and non-deductive logic, viewpoint on mathematics is also changing. It is just like the case that Newton Physics gives enough understanding to our macro-world, but Quantum Physics still works better in advanced case such as for micro-worlds or some specific cases for macro-world. Non-deductive logic or logical probability is a necessity for problems kind of Riemann Hypothesis. But, even now, we are not sure that it gives enough understanding about it or not. And probably, we will not be sure; because this will not be the last case.
[1] : Mathematical Induction and Induction in Mathematics, p.3, http://www.psych.northwestern.edu/~rips/documents/mathinduct3.pdf
[2] : http://www.matematikcafe.net/tumdengelim-tumevarim-tarihsel-gelisimi-t-3195.html
[3] : http://en.wikipedia.org/wiki/Riemann_hypothesis
[4] : James Franklin, Non-deductive Logic in Mathematics, Brit. J. Phil. Sd. 38 (1987), p.1
http://www.earlham.edu/~peters/courses/logsys/math-ind.htm
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