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3 ideas
| 16345 | That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction [Halbach] |
| Full Idea: The observation that Peano arithmetic is relatively interpretable in ZF set theory is taken by many philosophers to be a reduction of numbers to sets. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 23) | |
| A reaction: Nice! Being able to express something in a different language is not the same as a reduction. Back to the drawing board. What do you really mean by a reduction? If we model something, we don't 'reduce' it to the model. |
| 7953 | Reasoning needs to cut nature accurately at the joints [Plato] |
| Full Idea: In our reasoning we need a clear view of the ability to divide a genus into species, observing the natural joints, not mangling any of the parts, like an unskilful butcher. | |
| From: Plato (Phaedrus [c.366 BCE], 265d) | |
| A reaction: In modern times this Platonic idea has become the standard metaphor for realism. I endorse it. I think nature has joints, and we should hunt for them. There are natural sets. The joints may exist in abstract concepts, as well as in objects. |
| 16121 | I revere anyone who can discern a single thing that encompasses many things [Plato] |
| Full Idea: If I believe that someone is capable of discerning a single thing that is also by nature capable of encompassing many, I follow 'straight behind, in his footsteps, as if he were a god'. | |
| From: Plato (Phaedrus [c.366 BCE], 266b) | |
| A reaction: [Plato quote Odyssey 2.406] This is the sort of simple but profound general observation which only the early philosophers bothered to make, and no one comments on now. Encompassing many under one is the very essence of thinking. |