Types of reasoning in mathematics

Question

Can we say that analytical approaches is deductive reasoning and numerical approaches(numerical analysis) is inductive reasoning ?

Answer 1

Hilbert in his book, Geometry and the Imagination, pointed out that there were two modes of mathematical reasoning, one deductive and the other inductive. He considered the inductive process to be more important, but in fact the two are intimately involved in a kind of dialectic. Mathematics would not have got very far relying on just one.

Thus mathematical reasoning is a dialectic of inductive and deductive thinking.

Answer 2

Your linked pages are to mathematical analysis and numerical methods. Disclaimer: I am not an expert in either.

However, consider your typical numerical methods course. It is a study of algorithms used for approximations, maybe take newtons method. Does newtons method always work? (It doesn't, just consider derivative equal to 0, or function not continuously differentiable, etc...). So applying a numerical method really relies on some background conditions guaranteeing convergence to a solution ( at least, in a large class of cases). This generally takes the form of an analytic proof. So even the application of numerical methods is primarily deductive (if you consider proofs to be deductive). I take it that this is what @davidGudeman was trying to say.

Answer 3

Mathematics is all deductive, not inductive in the way those words are used in logic. There is something called mathematical induction, but that sort of reasoning is logical deductive reasoning not logical inductive reasoning according to most definitions.

Mathematics however does deal with inductive arguments in various ways. As you suggested, numerical approximation techniques are one such method. There are also probability, statistics, and fuzzy sets, that I know of. All are deductive methods to deal with problems that can't be perfectly solved deductively.

These methods use deduction to come up with an answer that is not exact or not complete, but that is better than no answer at all. You could say that the solution is inductive in the sense that you can't prove the solution you get is the true answer to the problem, but you did not arrive at that solution through inductive reasoning. If you used mathematics, you used deductive reasoning.

ADDENDUM: since a commenter seems not to have understood my point, let me try an example: you are going to draw two balls out of an urn without looking. After you draw the first ball, you will replace it and remix the urn before drawing the second. There are 10 balls in the urn, all identical to the touch, but 9 of them are black and one of them is white. The urn has been thoroughly mixed. You want to predict whether you are likely to draw the white ball. You make the following steps of reasoning:

1. On a single draw, there is a 0.1 chance of drawing a white ball.
2. The chance of drawing a white ball in two tries is 0.1+0.1-0.1*0.1=0.19.
3. This means that you would expect to see a white ball roughly 19/100 times you try the experiment.

Steps 1 and 3 of this reasoning are inductive. Step 2, the step where you do the actual mathematics, is deductive. There is inductive reasoning in the overall process, but the part of the reasoning that can be described as mathematics is deductive.

Mathematics is used all the time to support inductive reasoning. In fact, that's the main use of mathematics. Even in physics, if you use mathematics to predict the acceleration of a mass under a force, your overall process is inductive but the math part of the reasoning is deductive.