Timeline for Causality, if any, in mathematics itself
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Apr 7, 2023 at 23:33 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Added more detail
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| Apr 7, 2023 at 22:43 | comment | added | fedja | @LSpice they are defined to make what I say true. No objections then :lol: | |
| Apr 7, 2023 at 21:33 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Reformatted slightly for emphasis
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| Apr 7, 2023 at 20:35 | comment | added | LSpice | @fedja, re, with such (wise) sayings in mind I carefully avoided saying that we understand all the properties of the objects we are defining, only that we had captured all of them. It is doubtless true that not all the properties of the complex sine function are understood; but, to the extent that they are pure mathematical properties, that gap is due only to our failure fully to explore our model, not to the failure of the model itself. (If you worry about the definitions of the terms I use … they are defined to make what I say true. 😄) | |
| Apr 7, 2023 at 12:32 | comment | added | fedja | @LSpice "we can be totally sure we've captured all the relevant properties of an object.". There is an old story about Bernstein. A young (and also prominent) mathematician came to him and started to talk about the latest results in the theory of entire functions of exponential type. Bernstein listened for a while and then said "That is all beautiful, but for me there still is so much mysterious and not yet understood about the function $\sin z$!". | |
| Apr 5, 2023 at 22:46 | comment | added | Timothy Chow | @LSpice I had to look up the Free Dictionary to realize that by SEP you meant Shipbuilders' Estimating Package. :-) | |
| Apr 5, 2023 at 21:24 | comment | added | LSpice | @AlecRhea, re, that's why I specified pure mathematics! Applied mathematicians suffer, like everyone else, from the imperfect approximations of their models to reality. But I am interested in the representation theory of $p$-adic groups, and, though I don't know anywhere close to all that there is to know about them, I know for sure that my model fully and accurately reflects what, say, a reductive group is, because it's the definition of a reductive group. Whether that definition is useful, or accurate, in applications to reality is SEP …. | |
| Apr 5, 2023 at 19:21 | comment | added | Alec Rhea | @LSpice Au contraire; the issue is that the models they consider always necessarily miss some of the subtlety involved in reality, because everything they're trying to consider contains protons (amongst other putatively infinitely complex elementary particles). I dream of a world where we consider mathematical entities that actually correspond to the full breadth of complexity in reality, but would be amazed if we came close in my lifetime. | |
| Apr 5, 2023 at 15:18 | history | became hot network question | |||
| Apr 5, 2023 at 14:21 | comment | added | LSpice | @AlecRhea, re, I don't think it's so much that mathematical objects are simpler than "real" objects, though they doubtless are, as that we can be totally sure we've captured all the relevant properties of an object. That is, a physicist or biologist must always be concerned that their model is oversimplifying away some relevant portion of the problem, whereas pure mathematicians know for sure that we have modelled the problem completely accurately because the problem is defined to be what we modelled. | |
| Apr 5, 2023 at 13:30 | answer | added | Timothy Chow | timeline score: 12 | |
| Apr 5, 2023 at 13:06 | comment | added | Alec Rhea | @fedja Very true :^). I think waiting is fine, it seems a reasonable number of people are interested in the question whether or not we understand it. | |
| Apr 5, 2023 at 12:02 | comment | added | fedja | @AlecRhea Yeah, but that journey is quite individual. What causes me to turn right often causes you to turn left at the same intersection and vice versa so there doesn't seem to be any universal causation here either. :-) OK, let's wait until somebody comes with a reasonable answer (or should we vote to close?). | |
| Apr 5, 2023 at 9:04 | comment | added | Alec Rhea | @fedja I don't really understand the mathematical content of the question either -- this seems to be asking for a philosophical/experiential description of what causes our journey through the platonic landscape to take the shape it does. I'm not sure how to answer that question in a mathematical way. | |
| Apr 5, 2023 at 9:02 | comment | added | Alec Rhea | What you're calling 'rigor' might be better dubbed 'precision of thought', and sure, mathematics is completely precise thought. Rigor exists elsewhere, but it's difficult/potentially impossible to think completely precisely about 'reality' as physicists/biologists etc. try to do. We can think with complete precision about our subject because we choose to think about things that are far simpler than the whole of reality (or any individual piece of reality). | |
| Apr 5, 2023 at 6:04 | comment | added | Daniel Asimov | When I say mathematics I mean pure mathematics, and my point was only that, as I see it, that kind of rigor is what defines mathematics. Any other discipline that uses the same level of rigor is, by definition, mathematics (whatever else it may be). | |
| Apr 5, 2023 at 4:48 | comment | added | fedja | @AlecRhea May be he does, maybe he doesn't. The opinion is certainly too strong to be literally true, but I wouldn't call it completely ungrounded either. Anyway, I'm trying to understand the question as posted and fail. The answers so far somehow seem (to me) to miss the essence of it either... Do you understand what exactly the OP is looking for (from his edit, it is clear that it is certainly not just some kind of "modus ponens")? | |
| Apr 5, 2023 at 1:19 | comment | added | Daniel Asimov | Matt F. — I'm not sure I agree, because mathematics is the only discipline in which rigor even exists. | |
| Apr 5, 2023 at 1:07 | comment | added | user44143 | Is there a rigorous concept of causality that you like for physical objects or economic phenomena? Eg would you say the solsticial sun shines through Stonehenge because of its layout, or that the Stonehenge layout is because of the way the solsticial sun shines? Do you like Granger’s analysis of economic causation? If you don’t have a rigorous concept you like elsewhere, I doubt you’ll find one you like for mathematics. | |
| Apr 5, 2023 at 1:02 | comment | added | tox123 | Tangential, but FWIW there's somewhat of a notion of causality in statistics, where there is an entire field dedicated to causal inference. I wonder though if you could adapt categorical semantics for modality to construct a notion of causation? | |
| Apr 5, 2023 at 0:50 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Added detail
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| Apr 5, 2023 at 0:40 | comment | added | Alec Rhea | From a logical standpoint I think this is just syntactic entailment, right? That, or the notion of 'modeling' something in model theory -- in any event, I think you're looking for $\vdash$ or $\models$ as your causality relation. | |
| Apr 4, 2023 at 23:45 | comment | added | Sam Hopkins | Reverse mathematics is sort of like this: investigating which theorems imply which other theorems, in very weak background systems. | |
| Apr 4, 2023 at 23:39 | history | asked | Daniel Asimov | CC BY-SA 4.0 |