Timeline for The massive problem with regarding string manipulations as the foundation of mathematics
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2023 at 7:33 | comment | added | Paul Ross | @RyderRude, I think it's worth challenging this suggestion that there needs to be one established pre-mathematics. You could imagine a kind of convergent evolution scenario where multiple different species come to learn to communicate through incommensurable cognitive mechanisms, and one of the virtues of Formalism is that we can understand how they could nonetheless collaborate on mathematical models despite potentially very different systems of reasoning/logic. | |
| Mar 10, 2023 at 18:59 | history | became hot network question | |||
| Mar 10, 2023 at 18:31 | answer | added | Bumble | timeline score: 2 | |
| Mar 10, 2023 at 16:56 | comment | added | Ryder Rude | Let us continue this discussion in chat. | |
| Mar 10, 2023 at 16:54 | comment | added | Frank | @RyderRude But you said "in nature" - so you have to adduce some observation of nature, or experiment to make an argument about this being "never violated in nature". Asserting is not enough for a claim about nature. | |
| Mar 10, 2023 at 16:48 | comment | added | Ryder Rude | @Frank e.g. "not A AND A can't be true" is never violated. The underlying meta-language of string manipulation too has to adhere to this rule. e.g. "String B can be inferred from string A AND string B cannot be inferred from string A" is always false for the string manipulation system. Quantum Logic is merely a special case of string manipulation systems, so all of this applies to quantum logic too. | |
| Mar 10, 2023 at 16:20 | comment | added | Frank | @RyderRude "Standard logic is never violated in nature": what do you mean? Can you provide an example? | |
| Mar 10, 2023 at 15:57 | comment | added | Ryder Rude | @Frank Standard logic is never violated in nature. Quantum Logic and Fuzzy logic are physics models formulated within standard logic, with very specific applications. One can come up with arbitrary string manipulation rules, but one can never avoid the presence of standard logic in the meta-language of any string-manipulation system, even if it is automated. Standard logic is extremely elementary and it is born out of the notion of "collections" alone (which is another notion that can't be defined but only be discovererd) | |
| Mar 10, 2023 at 15:11 | comment | added | Frank | For those kinds of questions, the "decision making" doesn't matter - we (theoretically) reason on "all decisions" as in - "all possible string manipulations are carried out" (but nobody really carries them all out). | |
| Mar 10, 2023 at 15:09 | comment | added | Frank | I think also, there are nuances. To a mathematician, it is true that in principle you can derive theorems by string manipulations from axioms following syntactic rules. However, in practice, no mathematician works that way. It may be useful to think like that if you work on computability or provability where you may ask questions like "can this theorem be reached by string manipulations?" or "how efficient is it to find all theorems by string manipulations?" but it's a small sliver of mathematics. | |
| Mar 10, 2023 at 15:02 | comment | added | Frank | @RyderRude I think that when we say "game of string manipulation" what it really means is that the derivations can be automated, as Conifold points out. I don't think that recognizing that derivations follow strict rules that can be automated (at least in principle), is related to the foundations. For example, the rules would be in FOL, and the foundations would be ZF or NBG, but both would presumably use FOL to conduct derivations. | |
| Mar 10, 2023 at 14:58 | comment | added | Frank | @RyderRude Standard logical rules are certainly not discovered to hold in nature. We can alter them and still obtain formal systems in which we can derive strings from other strings. I think there is a confusion between syntax and semantics here. | |
| Mar 10, 2023 at 8:57 | comment | added | Ryder Rude | @PaulRoss It's fine if they're defining mathematics that way. But for any discussion of foundations to be interesting, one should discuss what they are defining to be "pre-mathematics". Animals too can be seen to understand the concepts of (if, then, and, or, not). I believe that standard logical rules are discovered and are fundamental to existence. The language that humans have developed only tries to capture this underlying logic. | |
| Mar 10, 2023 at 7:36 | comment | added | Paul Ross | It's not a problem for formalists to say that the fundamental type of decision making in mathematics is premathematical. The point of maths, for a formalist, is providing contextual structure and mathematical sophistication to communication. If human communication protocols weren't at least in the most part sound and consistent, we wouldn't have gotten to the stage of being able to build formal mathematical models anyway. This also meshes with a pluralist approach to logic - there may not be one single correct set of logical communication protocols, which extends to a plurality of maths-es. | |
| Mar 10, 2023 at 7:26 | comment | added | Conifold | Ideas of logic do not underlie string manipulations themselves. Rules can be programmed into a machine and it will follow them, with no ideas at all. But there are logical (and also arithmetical and set-theoretic) ideas in reasoning about string manipulation because their syntax is no simpler than arithmetic. This is what SEP calls "metatheory problem" of game formalism. There are no good answers to it so far, and so it remains "a position which it is fair to say most philosophers of mathematics still think hopeless". | |
| Mar 10, 2023 at 7:05 | comment | added | Hudjefa | We must talk more to(?) mad folks. | |
| Mar 10, 2023 at 6:47 | comment | added | Mauro ALLEGRANZA | "Formalists believe that mathematics is just a game of string manipulation". Not exactly; see e.g. Curry. | |
| Mar 10, 2023 at 6:33 | answer | added | kutschkem | timeline score: 2 | |
| Mar 10, 2023 at 5:36 | history | edited | Ryder Rude | CC BY-SA 4.0 |
added 2 characters in body
|
| Mar 10, 2023 at 4:43 | history | edited | Ryder Rude | CC BY-SA 4.0 |
deleted 3 characters in body
|
| Mar 10, 2023 at 4:32 | history | asked | Ryder Rude | CC BY-SA 4.0 |