The purpose of heuristic reasoning is to discover the solution to the present problem. Heuristic reasoning as an approach in that regard is not a final and strict approach but a provisional and plausible approach to the problem. We can only achieve complete certainty when we obtain the complete solution.
At UnicMinds, we are very keen in developing intuition of the concept in students so that students can use that intuition to develop their own methods and theories like heuristic reasoning. This method helps the students to build their own ways of proving things if not through a mathematically rigorous method. But, this foundation of intuition can then be used as a synthesis of the concept to build a solid rigorous way of solving the problem and why it is done that way.
The teaching of certain subjects such as calculus can take advantage of this heuristic way of thinking, but both its advantages and disadvantages are to be recognized. Text books would also do a world of good to show all the heuristic arguments leading to the invention of a mathematical concept and its proof. Often mathematicians build from an intuition with heuristics towards various proofs.
If you cannot solve the proposed problem, don’t let it affect you too much but try to find consolation in an easier success. Try to solve a related problem, and then you may find courage to attack your original problem again. The spirit of a human being is the greatest in going around an obstacle that cannot be overcome directly. Invent a related problem and try to solve that problem, and that will give you some solace.
Means to Solve Problems
Mathematically speaking, there are different ways to solve a problem: generalization, induction, specialization, analogy, decomposition, combinations and recombinations, and more. Normally, the easier a problem is, the easier it is to solve it. However, it is counter-intuitive that many mathematicians and inventors historically wouldn’t have picked the problem they picked had they known its difficulty in time before. So, most of the time, it works that an ambitious plan doesn’t mean less chances of success. It too has high levels of success. Obviously, the ambitious plan shouldn’t be dependent on some pretension but on some vision of things beyond those immediately present.
The other thing is to not try to bite the problem all at once. Do not look at the problem in its entirety but focus on the unknowns. Look at the problem schematically and focus on the unknowns. See, if there are previous problems that had a similar unknown and how those previous problems were solved. If we foresee that the unknowns can be solved in some way, then we have made essential progress. But, sometimes we will not have any previous problem with the same kind of unknowns. For example, the problem is to find the surface area of a sphere with a given radius. When Archimedes was solving this problem, he didn’t know of any previous problem with the same unknown. However, there were similar problems that Archimedes studied, like that of the surface area of a cylinder. Solving for the surface area of a sphere is much more complex than solving for the surface area of a cylinder. In fact, he did approximate a sphere to two cones in his approach.
Often, we think like this in the real world too. We think if you could get the surface area of a cylinder then the surface area of a sphere is just an extension of the same solution. But, it isn’t. But, we usually have a good chance of solving something when you recollect similar problems with similar unknowns and try to solve using those methods.
Provisionally, heuristic reasoning is important in discovering the solution, but you shouldn’t take it for a proof. You must guess, but also examine your guess. It is therefore emphasized that all sorts of problems and even puzzles are within the scope of heuristic.
Hope you liked this post, thank you.
You may like to read: Why Do We Need Complex Numbers in Life?, Critical Thinking for Juniors, & ANNs vs. CNNs vs. RNNs vs. Transformers
