A Quote by Karl Popper
Now this principle of induction cannot be a purely logical truth like a tautology or an analytic statement. . . .
Related Quotes
Now ... the basic principle of modern mathematics is to achieve a complete fusion [of] 'geometric' and 'analytic' ideas.
Science is a principle and a process of seeking truth. Truth cannot be purchased, and thus, truth cannot be altered by money.
Science sometimes improves hypotheses and sometimes disproves them. But proof would be another matter and perhaps never occurs except in the realms of totally abstract tautology. We can sometimes say that if such and such abstract suppositions or postulates are given, then such and such abstract suppositions or postulates are given, then such and such must follow absolutely. But the truth about what can be perceived or arrived at by induction from perception is something else again.
However logical our induction, the end of the thread is fastened upon the assurance of faith.
If we have no idea why a statement is true, we can still prove it by induction.
If you have a correct statement, then the opposite of a correct statement is of course an incorrect statement, a wrong statement. But when you have a deep truth, then the opposite of a deep truth may again be a deep truth.
It has to be simple, but then you deliver them a principle: The simple truth is, as a matter of principle, we cannot spend more than we take in. Something - that changes the tone of the debate.
To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.
The question of relevance comes before that of truth, because to ask whether a statement is true or false presupposes that it is relevant (so that to try to assert the truth or falsity of an irrelevant statement is a form of confusion).
Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. "I don't have any whiskey," may be a fact but it is not a truth.
The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
Eyes blinded by the fog of things cannot see truth. Ears deafened by the din of things cannot hear truth. Brains bewildered by the whirl of things cannot think truth. Hearts deadened by the weight of things cannot feel truth. Throats choked by the dust of things cannot speak truth.
It is time, therefore, to abandon the superstition that natural science cannot be regarded as logically respectable until philosophers have solved the problem of induction. The problem of induction is, roughly speaking, the problem of finding a way to prove that certain empirical generalizations which are derived from past experience will hold good also in the future.
A tautology's truth is certain, a proposition's possible, a contradiction's impossible.
Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.
"Who are we to say what is right and what is wrong?" is the common refrain under the doctrine of pure pluralism. Clearly, society cannot long survive if this principle is pushed to its logical conclusion and everyone is free to write his own laws.
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