In this post, I share one of my favorite passages ever written about mathematical thinking and problem solving. This is George Pulya’s definition of the inductive attitude. Enjoy!
“In our personal life we often cling to illusions. That is, we do not dare to examine certain beliefs which could be easily contradicted by experience, because we are afraid of upsetting our emotional balance.
There may be circumstances in which it is not unwise to cling to illusions, but in science we need a very different attitude, the inductive attitude.
This attitude aims at adapting our beliefs to our experience as efficiently as possible. It requires a certain preference for what is matter of fact. It requires a ready ascent from observation to generalizations, and a ready descent from the highest generalizations to the most concrete observations. It requires saying “maybe” and “perhaps” in a thousand different shades. It requires many other things, especially the following three:
1. Intellectual courage: we should be ready to revise any one of our beliefs.
2. Intellectual honesty: we should change a belief when there is a compelling reason to change it.
3. Wise restraint: we should not change a belief wantonly, without some good reason.
These points sound pretty trivial. Yet one needs rather unusual qualities to live up to them.
The first point needs intellectual courage. You need courage to revise your beliefs. Galileo, challenging the prejudice of his contemporaries and the authority of Aristotle, is a great example of intellectual courage.
The second point needs intellectual honesty. To stick to my conjecture that has been clearly contradicted by experience just because it is my conjecture would be dishonest.
The third point needs wise restraint. To change a belief without serious examination, just for the sake of fashion, for example, would be foolish. Yet, we have neither the time nor the strength to examine seriously all our beliefs. Therefore it is wise to reserve the day’s work, our questions, and our active doubts for such beliefs as we can reasonably expect to amend. ‘Do not believe anything, but question only what is worth questioning.’
Intellectual courage, intellectual honesty, and wise restraint are the moral qualities of the scientist.”
- Mathematics and Plausible Reasoning:
Volume 1: Induction and Analogy in Mathematics
by George Pulya, page 7 – 8
- As you read the passage above, think about how you might apply the inductive attitude to any subject you are learning, not just mathematics. What comes up for you as you think about this?
- Read any of George Pulya’s books: How to Solve It, Mathematics and Plausible Reasoning (Vol 1 & 2), or Mathematical Discovery (Vol 1 and 2). As you read, come back to the inductive attitude. How do the techniques that Pulya is guiding you to develop related to the inductive attitude? How would you edit this passage to reflect your learning?
- How is the inductive attitude related to the scientific method? How are these ideas similar and how are they different? What inter-dependencies exist between the inductive attitude and the scientific method? Do you need one to do the other? Can either exist without each other?
- I believe that artists, writers, authors, historians, politicians, and almost anyone who learns for a living uses some key principles from this inductive attitude, regardless of whether or not they describe their work with these words. In other words, mathematicians don’t have a monopoly on these tools. Choose any non-STEM field: say like art. How might the inductive attitude show up in the lives of people who work in the field you chose?
- Let me go a little further. I think that human beings are sophisticated learners by our very nature. We might not be able to say describe in word what is happening in our brains as we learn. But part of what it means to be human is to learn, to be curious, and to seek to understand our world. I also believe it is in our nature to seek out challenging tasks, to work towards mastery, and to engage in meaningful work. In other words, I claim that the inductive attitude put into words fundamentally human characteristics. Refute or substantiate this claim for yourself.