If we see green apples fall from a tree every day, why should we reason inductively and conclude that every apple that falls from the tree will be green? Why not conclude instead that the next apple to fall from the tree will be red?

via Induction: The Rocky Foundations of Science — The Platonic Realm

I wanted to share this post by Matthew Davies as it was a useful explanation of the distinction between deductive and inductive reasoning. In particular, with deductive reasoning we start with assumptions of a general, then we determine what can be logically inferred in specific situation. On the other hand, in with inductive reasoning, we make observations of particular situations, then extend this observed relations to make conclusions about the general case.

While the author of the blog provides his own examples, I will provide a simple one. Suppose I define a sequence by giving the first two terms a_{1}=1, a_{2}=1, and a_{n}=a_{n-1} +a_{n-2} for all n>2 (note that this is the Fibonacci sequence), then we can find all terms of the sequence by using this rule. Therefore, the first 5 terms would be 1,1,2,3,5. Note here that we **know** that are terms are correct because we are using logical inference and definitions to find the answers.

On the other had, suppose that you were given the terms a_{1}=1, a_{2}=1,a_{3}=2, and a_{4}=3. You could then inductively come to the conclusion that we have the same sequence as previously defined. However, do you know that this is the case? As I provide you more terms in the sequence you may be more comfortable in your conclusion; however, you still cannot be sure that your answer will always be correct. For example, I could have the sequence agree for the next 100 terms, then be 0 for all following terms.

While the author stops at giving the distinction between the two, I would like to take the further step of suggesting that using both types of reasoning in concert will often lead to the best results. For example, if we look at the population of pairs of bunnies, starting with one pair, we will indeed see that for the first 5 months, the pairs will be 1,1,2,3 and 5. Again, if we make further observation, this pattern will indeed continue, so we could make the conjecture that the population will follow Fibonacci sequence. However, we are provided no explanation as to why this may be the case, so we are left wondering if the model will fall apart.

If we now make the further observation that Fibonacci did, we realize that if we start with one pair of bunnies, it will take them a month to mature before reproducing. Then, they will produce a new pair of bunnies every month after the first. In the same way, their offspring will take a month to mature, but then will continue to reproduce a new pair of bunnies every month. Because of this, after the first two months, the new population will be the sum of the population from the previous month (as these bunnies are still there) and the population from the month prior to this (as these are all pairs of bunnies from the now mature pairs). We have now come up with a rule that describes the population of bunnies so we know that, under our assumptions, the pattern will continue indefinitely.

That is, we quite often can find the patterns by using an inductive argument. However, this does not provide justification for the pattern continuing. Hence, we need to find a deductive argument to show that our observations are correct. In this way, we can both find and prove that patterns exist. As such, I would provide the following answer to Matthew’s question Is Everything Mathematics? Yes, everything is mathematics, in so much as mathematics is the art of logic and reasoning. Without the ability to provide arguments and determine their validity there would be no way of forming thoughts on, models of, or descriptions of any phenomena. However, mathematics explains why things occur with given assumptions. It is therefore the responsibility of the applied researcher to show that the situation they are looking at in fact satisfies the assumptions they are making and to determine if the assumptions will hold in general.

As a personal observation I’d like to share with educators, I find that I get the best retention from a lecture or lesson if I first provide an inductive argument to get the students to recognize what is going on, followed by a deductive argument showing that the pattern will always work.

I enjoy the posts Matthew’s made on The Platonic Realm and I hope you take the time to check it out. Indeed, he attempts to answer the question of what assumptions can be made within our universe in an entertaining way. Let me know what you think about logic and assumptions below. Also follow my blog for updates on this website.