Timeline for Defining the standard model of PA so that a space alien could understand
Current License: CC BY-SA 4.0
14 events
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| Nov 18, 2023 at 18:35 | comment | added | Mikhail Katz | Thanks, @Joel. In fact, your comment is possibly the best answer to this question. Why don't you format it as an answer to give it more visibility? | |
| Nov 17, 2023 at 13:07 | comment | added | Joel David Hamkins | Mikhail, thanks for posting a link to my essay, which is exactly about this topic. In my view, we don't ultimately have a self-contained logically satisfactory definition the finite. People say, 1, 2, 3, and so on, but of course that is inadequate. The best definitions, such as the Dedekind categoricity argument, all rely fundamentally on induction, which requires a background conception of set theory or higher-order logic. Clearly we cannot ground the definiteness of the natural number concept on these higher concepts, whose definiteness is surely only less secure. How then are we to do it? | |
| Nov 17, 2023 at 11:37 | comment | added | Mikhail Katz | @Burak, in fact Leibniz considered such a "set" to be a contradictory notion (as explained in my answer), whereas infinitesimals are merely fictional (not contradictory). | |
| Nov 17, 2023 at 11:32 | comment | added | Burak | @SamHopkins: Counting numbers is easy, a child can do that. But understanding the smallest collection of objects that contains $0$ and $S$ is difficult, even for professional mathematicians, as shown by numerous discussions on MO. So my comment still stands that the set of natural numbers is no easier to understand than imagining points on a line that are infinitely close to $0$. | |
| Nov 17, 2023 at 11:16 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Nov 17, 2023 at 11:15 | comment | added | Mikhail Katz | Sam Hopkins, you wrote: "Of course N is more basic than Robinson's infinitesimals" as if somebody disagrees with you. Perhaps you meant is as a response to @Burak's comment, but your comment came out as an (unjustified) criticism of my answer. Leibniz would have been the first to agree that infinitesimals are more complicated than 1,2,3,... Indeed, the counting numbers are assignable, whereas infinitesimals are inassignable, in Leibniz's terminology. | |
| Nov 16, 2023 at 20:47 | comment | added | Rob Arthan | @SamHopkins: I think anecdotal and anthropological evidence is against you. "Understanding what strings are" is prior to counting: a child who asks her father "what does two mean?" has a grasp of grammar but not of arithmetic. The Pirahã people apparently speak a complex language but are wilfully against abstractions such as numbers. Mikhail's post seems to be to be very pertinent. | |
| Nov 16, 2023 at 16:29 | comment | added | Sam Hopkins | Of course $\mathbb{N}$ is more basic than Robinson's infinitesimals. The whole discussion of Peano Arithmetic and first order logic here is a red herring. Imagine someone learning logic for the first time: they have to understand what strings are, and how to manipulate them. These ideas are at least as sophisticated as the counting numbers. Thurston's "On proof and progress in mathematics" (arxiv.org/abs/math/9404236) gives a much better account of the way mathematical ideas are actually developed and conveyed than what is described by any axiomatic approach. | |
| Nov 16, 2023 at 12:56 | comment | added | Mikhail Katz | Currently there are three downvotes on this question. I would appreciate constructive input from critics. | |
| Nov 16, 2023 at 10:06 | comment | added | Burak | @SamHopkins: While I see why you think it seems bizarre, here is a more advanced analogue that may explain why it may not be bizarre: Mathematicians of a certain era thought that infinitesimals are problematic and they avoided using them. Then came Robinson... One may say that Robinson would have hard time communicating the idea of consistently doing calculus with infinitesimals to, say, Bishop Berkeley. The only difference here is that $\mathbb{N}$ seems more "basic" but that's just a habit of thinking. Is it really? | |
| Nov 16, 2023 at 9:39 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Nov 15, 2023 at 13:34 | comment | added | Mikhail Katz | @SamHopkins, what we can successfully commucate to others are naive counting numbers, or at best what logicians would refer to as metalanguage natural numbers. "Standard models" are another matter, that would have been rejected by Leibniz for one. | |
| Nov 15, 2023 at 13:26 | comment | added | Sam Hopkins | Irrespective of any deep philosophical musings, in practice we clearly communicate the idea of numbers to other humans all the time. It seems bizarre to think there is a group of humans who we couldn't, with some effort, communicate the idea of numbers to. | |
| Nov 14, 2023 at 14:19 | history | answered | Mikhail Katz | CC BY-SA 4.0 |