What's the basic difference between induction and deduction in terms of
geometric proofs?
Hey rapture.
Deduction starts with lots of information and tries to combine the different relations and predicates in ways to arrive at whether a statement is considered true or false. If uncertainty is involved, then we look at whether something is likely or unlikely to be true rather than if it is absolutely true (or not).
Induction starts with a statement that applies to a small number of outcomes and tries to see if it can be applied to a much larger number of outcomes. So in normal induction you start with a single case and then you try and prove that it works for many other cases. This is not the only type of induction - but the idea behind it is shared with other kinds of induction.
So in summary - deduction takes complete information about something and tries to combine pieces of information and relations to prove true/false or likely/unlikely with regard to some statement while induction starts with a statement that applies to a small number of things and tries to see if it applies to many more things in some way - like with proof by induction.
Technically, "mathematical induction" is both a process of induction and a process of deduction.
Even though the proof of the statement itself will be a process of deduction, it also requires you to actually have found a pattern that you can try to prove. This is an inductive process.
However, you may be thinking of "inductive reasoning" (Deductive Reasoning vs. Inductive Reasoning) which is not a mathematical method of proof at all!
In epistemology, an inductive argument is the argument that if A is true for every member of a large enough sample from some class, then A is almost certainly true for every member of the class. Practical life would be impossible without inductive arguments; everyone uses such arguments all the time. But it is important to remember that an inductive argument always leads to a conclusion that is NEVER certainly true. For example, "3 is prime, 5 is prime, 7 is prime, therefore all odd numbers are prime" is an inductive argument and is false. Consequently, what is called in epistemology an inductive argument is not considered a formal proof in mathematics.
Using the language of epistemology, "mathematical induction" is simply a form of deductive proof, and is in no way, shape, or form an inductive argument.
It is confusing terminology that "mathematical induction" is a form of what is called in epistemology a "deductive argument." What makes it doubly confusing is that conclusions proved by mathematical deduction, a deductive argument, are often initially hypothesized as a result of inductive experimentation. This is analogous to Popper's distinction between the psychology of scientific discovery and the logic of scientific discovery. I may do some mathematical experimentation and find a mathematical proposition that appears to be true (that is an inductive psychological process). I may then use mathematical induction to prove that proposition to be true (that is a deductive logical argument). Worse terminology can hardly be imagined.