Timeline for "Impredicative" definitions in mathematics
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Feb 20, 2023 at 18:50 | comment | added | J D | All I can tell you is I took a look at my real and complex analysis textbooks, and impredicativty is part and parcel of the axiomatic method. | |
| Feb 20, 2023 at 18:47 | comment | added | J D | At least reviewing SEP's Definitions. One's philosophy of definition is very much impacted by one's view on the realist-anti-realist debate, btw. If you can articulate whether you're an eliminative materialist, a subjective idealist, a scientific realist and so on, then you have a pre-existing basis for framing the nature of definition. | |
| Feb 20, 2023 at 18:46 | comment | added | J D | @Frank Of course not. That she can clearly determine the infimum in a set of lower bounds simply through the application of maximum goes to illustrate that the existence of the infimum, even sans label, means the infimum exists independent of merely assigning a label. Radical realism, Frank, is the belief that there is a structure to the universe independent of human beings and their language. The fact that you think there are "mathematical" definitions that are somehow fundamentally different in their psychological character than those of, say science, suggests you'd profit from... | |
| Feb 19, 2023 at 22:10 | comment | added | Frank | "My 7 year old can pick out the highest number in a set of numbers" - this is not an infimum. | |
| Feb 19, 2023 at 22:09 | comment | added | Frank | How about definitions in mathematics, which was the scope of this question? | |
| Feb 19, 2023 at 20:27 | comment | added | J D | I've gone back and revised your interpreted rewrite, and it sounds goofy in natural language because to explicitly state a premise which is presumed in the definition is redundant. I don't see it as impacting the analysis. Good luck. I hope you find a resolution! | |
| Feb 19, 2023 at 20:23 | comment | added | J D | its existence. Thus, no explicit language is required. If you think that an infimum exists, then you are merely articulating language to be sure that any claims about it are rigorously methodological. Any school child knows an infimum exists if presented with a set of numbers even if they don't know it's called an infimum. My 7 year old can pick out the highest number in a set of numbers. | |
| Feb 19, 2023 at 20:21 | comment | added | J D | of inner planets reclassifying Pluto as a dwarf planet? Of course not. It's just a revision of the convention of language used in the language-game, to paraphrase late Ludwig. Real definitions presume existential quantification in their articulation precisely because they are implicitly make the claim that what is discussed is real. This is a tenant of realism, mathematical or otherwise. The traditional view of nominalism stands in opposition, and that's Robinson's view on definitions: that strictly speaking, if something is real, crafting language to describe its structure PRESUPPOSES... | |
| Feb 19, 2023 at 20:18 | comment | added | J D | @Frank Well, Frank, that's a question about the metaphysics of definitions, to which the received view is that word-thing definitions like stipulative and lexical definitions, the definiendum and the definiens, have nothing to do with claims of existence, they are merely linguistic acts to associate one set of mention with another according to the use-mention dichotomy. Does a lexicographer claim Santa Claus exists because she publishes a definition of Santa? Of course not. Does the IAU claim anything about the existence of Pluto and inner planets by demoting through revised extension... | |
| Feb 19, 2023 at 17:19 | comment | added | Frank | What is a definition if it is not a claim of existence? | |
| Feb 19, 2023 at 17:08 | comment | added | Frank | Existence is central to impredicativity. The problem with a self-referential presentation is that you don't know if the object you are proposing even exists. | |
| Feb 19, 2023 at 17:02 | comment | added | Frank | Your proposed "Frank's rewrite" is not accurate. In particular, "Frank's actual rewrite" carefully includes existence premises. You can't just remove those. | |
| Feb 19, 2023 at 16:46 | history | edited | J D | CC BY-SA 4.0 |
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| Feb 19, 2023 at 16:39 | history | edited | J D | CC BY-SA 4.0 |
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| Feb 19, 2023 at 16:32 | comment | added | J D | If you don't buy this post, then my gas tank is empty, but either way, thanks for posting this question! It's a fascinating topic. :D | |
| Feb 19, 2023 at 16:31 | history | answered | J D | CC BY-SA 4.0 |