Tom Parson's List: Knowledge

• Where the empiricist encounters difficulty is with the truths of formal logic and mathematics, for they appear to everyone to be necessary and certain.
• The empiricist must deal with these truths in one of two ways: either say that they are not necessary truth, and then account for the universal conviction that they are; or say that they have no factual content, and then explain how a proposition without factual content can be true and useful and surprising.
• He maintained that these propositions were inductive generalizations based on an extremely large number of instances. I do not think that this solution is acceptable.

• I found instead that they only amounted to nine. One would not say that the mathematical proposition '2x5=10' had been confuted. One would say that I was wrong in supposing that there were five pairs to start with, or that one object had been taken away while I was counting, or that two of them had coalesced, or that I had counted wrongly.

    

  To take another example: if what appears to be a Euclidean triangle is found by measurement not to have angles totalling 180 degrees, we do not say that we have met with an instance which invalidates mathematical laws. We say that we have measured wrongly or that the triangle is not Euclidean.
• 'My friend has not stopped writing to me' is not, as it appears to be, contradictory to 'My friend has stopped writing to me', only contrary to it. For it means 'My friend wrote to me in the past, and he still writes to me'. Thus we preserve the law of the excluded middle by showing that the negating of a sentence does not always yield the contradictory of the proposition originally expressed.
• Whatever instance we take, the principles of logic and mathematics are true universally simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves. In other words, the truths of logic and mathematics are analytic propositions or tautologies. This is a controversial statement.
• I think we can avoid Kant's confusions if we say that a proposition is analytic when its validity depends solely on the definitions of the symbols it contains, and synthetic when its validity is determined by the facts of experience.
• They call attention to linguistic usages and reveal unsuspected implications in our beliefs.
• But it would not provide new knowledge in the way that evidence that May Queens had been forbidden by law would provide new knowledge.
• A point not sufficiently brought out by Russell is that every logical proposition is valid in its own right. Its validity does not depend on it being incorporated into a system, that is to say, it being deducible from other analytic propositions. This is our justification for disregarding the question whether the propositions of mathematics are reducible to propositions of formal logic, in the way that Russell supposed. Mathematics may form a special class of analytic propositions containing special terms, but is analytic all the same.
• Geometry is a purely logical system, its propositions purely analytic, its diagrams not essential to completely rigorous geometry. The fact that most of us need the help of actual diagrammatic examples to be made aware of the consequences of geometric applications shows merely that our intellects are unequal to the task of carrying out very abstract processes of reasoning without the assistance of intuition.
• Our knowledge that no observation can ever confute the proposition '7+5=12' depends simply on the fact that the symbolic expression '7+5' is synonymous with '12'. And the same explanation holds good for every a priori truth.
• Thus we have shown that the truths of pure reason, the propositions which we know to be valid independently of all experience, are so only in virtue of their lack of factual content. To say that a proposition is true a priori is to say that it is a tautology

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