display all the ideas for this combination of philosophers
4 ideas
| 14776 | That two two-eyed people must have four eyes is a statement about numbers, not a fact [Peirce] |
| Full Idea: To say that 'if' there are two persons and each person has two eyes there 'will be' four eyes is not a statement of fact, but a statement about the system of numbers which is our own creation. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) | |
| A reaction: One eye for each arm of the people is certainly a fact. Frege uses this equivalence to build numbers. I think Peirce is wrong. If it is not a fact that these people have four eyes, I don't know what 'four' means. It's being two pairs is also a fact. |
| 14788 | Mathematics is close to logic, but is even more abstract [Peirce] |
| Full Idea: The whole of the theory of numbers belongs to logic; or rather, it would do so, were it not, as pure mathematics, pre-logical, that is, even more abstract than logic. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], IV) | |
| A reaction: Peirce seems to flirt with logicism, but rejects in favour of some subtler relationship. I just don't believe that numbers are purely logical entities. |
| 8473 | The logicists held that is-a-member-of is a logical constant, making set theory part of logic [Orenstein] |
| Full Idea: The question to be posed is whether is-a-member-of should be considered a logical constant, that is, does logic include set theory. Frege, Russell and Whitehead held that it did. | |
| From: Alex Orenstein (W.V. Quine [2002], Ch.5) | |
| A reaction: This is obviously the key element in the logicist programme. The objection seems to be that while first-order logic is consistent and complete, set theory is not at all like that, and so is part of a different world. |
| 19226 | We now know that mathematics only studies hypotheses, not facts [Peirce] |
| Full Idea: It did not become clear to mathematicians before modern times that they study nothing but hypotheses without as pure mathematicians caring at all how the actual facts may be. | |
| From: Charles Sanders Peirce (Reasoning and the Logic of Things [1898], I) | |
| A reaction: 'Modern' here is 1898. As a logical principle this would seem to qualify as 'if-thenism' (see alphabetical themes). It's modern descendant might be modal structuralism (see Geoffrey Hellman). It take maths to be hypotheses abstracted from experience. |