Most commented posts
- Induction, philosophy’s toughest zombie — 3 comments
- Pinker on Intelligence — 2 comments
- Getting nearer to the truth — 2 comments
- Why we need to disagree more — 2 comments
- The power of logic — 2 comments
Peter Monnerjahn
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The key to science
In general we look for a new law by the following process. First we guess it. Then we compute the consequences of the guess to see what would be implied if this law that we guessed is right. Then we compare the result of the computation to nature, with experiment or experience, compare it directly with observation, to see if it works. If it disagrees with experiment it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is — if it disagrees with experiment it is wrong. That is all there is to it. [156]
Measuring student progress
After teaching 10 years, the only good measure of student progress I know is the number of open problems they can successfully characterize.
The right not to tolerate the intolerant
Less well known is the paradox of tolerance: Unlimited tolerance must lead to the disappearance of tolerance. If we extend unlimited tolerance even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them.—In this formulation, I do not imply, for instance, that we should always suppress the utterance of intolerant philosophies; as long as we can counter them by rational argument and keep them in check by public opinion, suppression would certainly be unwise. But we should claim the right to suppress them if necessary even by force; for it may easily turn out that they are not prepared to meet us on the level of rational argument, but begin by denouncing all argument; they may forbid their followers to listen to rational argument, because it is deceptive, and teach them to answer arguments by the use of their fists or pistols. We should therefore claim, in the name of tolerance, the right not to tolerate the intolerant. We should claim that any movement preaching intolerance places itself outside the law, and we should consider incitement to intolerance and persecution as criminal, in the same way as we should consider incitement to murder, or to kidnapping, or to the revival of the slave trade, as criminal. [ch. 7, n4]
What counts in the future
Perhaps the greatest of all pedagogical fallacies is the notion that a person learns only the particular thing he is studying at the time. Collateral learning in the way of formation of enduring attitudes, of likes and dislikes, may be and often is much more important than the spelling lesson or lesson in geography or history that is learned. For these attitudes are fundamentally what count in the future. The most important attitude that can be formed is that of desire to go on learning. If impetus in this direction is weakened instead of being intensified, something much more than mere lack of preparation takes place. The pupil is actually robbed of native capacities which otherwise would enable him to cope with the circumstances that he meets in the course of his life. We often see persons who have had little schooling and in whose case the absence of set schooling proves to be a positive asset. They have at least retained their native common sense and power of judgment, and its exercise in the actual conditions of living has given them the precious gift of the ability to learn from the experiences they have. What avail is it to win prescribed amounts of information about geography and history, to win the ability to read and write, if in the process the individual loses his own soul; loses his appreciation of things worthwhile, of the values to which these things are relative; if he loses the desire to apply what he has learned and above all, loses the ability to extract meaning from his future experiences as they occur? [48-9]
Creative breakthroughs
What characterizes creative thinking, apart from the intensity of the interest in the problem, seems to me often the ability to break through the limits of the range—or to vary the range—from which a less creative thinker selects his trials. This ability, which clearly is a critical ablitiy, may be described as critical imagination. It is often the result of culture clash, that is, a clash between ideas, or frameworks of ideas. Such a clash may help us to break through the ordinary bounds of our imagination. [47]
The power of logic
Most people think that the purpose of an argument is to justify its conclusion — and to thereby establish its certainty — and that the problem with inductive arguments is that they fail to establish their conclusions with objective certainty since their conclusions may be false even if all of their premises are true. But this entirely confuses the issue, and it has even enabled inductivists to take the high ground in the debate, arguing that objective certainty is an impossible dream, and that inductive arguments are not, as a consequence, at fault for failing to achieve it.
But if the uncertainty of their conclusions were the problem with inductive arguments, then we would also have a similar problem with deductive arguments. For the premises of a deductive argument may be false. And the conclusion of a deductive argument may be false as well.
Contrary to what most people think, logical arguments cannot establish the truth, let alone the certainty, of their conclusions. And so contrary to what most people think, the problem with inductive arguments has nothing to do with the uncertainty of their conclusions.
The best that a logical argument can do is test the truth of a statement. But it can do this only by showing that its falsity is inconsistent with the truth of other statements that can only be tested and never proved. Our so-called ‘proof’ methods are really techniques for testing consistency. And the demonstration that a ‘conclusion’ follows from a ‘premise’ shows only that the falsity of the ‘conclusion’ is inconsistent with the truth of the ‘premise.’
That is all that is involved.
But so long as we regard contradictions as unacceptable, it is really quite a lot.
The inconsistency that marks a valid deductive argument — the inconsistency, that is, between the truth of is premises and the falsity of its conclusion — cannot force us to accept the truth of any belief. But it can force us, if we want to avoid contradicting ourselves, to reexamine our beliefs, and to choose between the truth of some beliefs and the falsity of others — because the falsity of the conclusion of a valid argument is inconsistent with the truth of its premises. …
And this is just another way of saying that what we call a proof actually presents us with the choice between accepting its conclusion and rejecting its premises. …
We construct logical arguments in order to persuade others of our beliefs. But the best we can do is to clarify a choice that they have to make. Inductive arguments, however, cannot even do this. [86-7]
Theirs not to reason why
The traditional scheme is, in essence, one of imposition from above and from outside. It imposes adult standards, subject-matter, and methods upon those who are only growing slowly toward maturity. The gap is so great that the required subject-matter, the methods of learning and of behaving are foreign to the existing capacities of the young. They are beyond the reach of the experience the young learners already possess. Consequently, they must be imposed; even though good teachers will use devices of art to cover up the imposition so as to relieve it of obviously brutal features.
But the gulf between the mature or adult products and the experience and abilities of the young is so wide that the very situation forbids much active participation by pupils in the development of what is taught. Theirs is to do—and learn, as it was the part of the six hundred to do and die. Learning here means acquisition of what already is incorporated in books and in the heads of the elders. Moreover, that which is taught is thought of as essentially static. It is taught as a finished product, with little regard either to the ways in which it was originally built up or to changes that will surely occur in the future. It is to a large extent the cultural product of societies that assumed the future would be much like the past, and yet it is used as educational food in a society where change is the rule, not the exception. [18-9]
How to train your chimpanzee
The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1−35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question—that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them).
A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?
I can understand the idea of training students to master certain techniques—I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process—that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much. [40-2]
The art of explanation
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity—to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes. [29]
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