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Are there thoughts that different physical multiverses or black holes have different math, I.d. physical events follow the mathematics that is not discovered yet, whose logic may be different from current first order logic, nonclassical logics or any other logics that admit already known algebraic (boolean and different) semantics?

TomR
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    Possibly relevant is my answer to How do I convince someone that $1+1=2$ may not necessarily be true?. – Dave L. Renfro Feb 18 '19 at 19:22
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    The objects being examined by first-order logic are not actually physical; they are intangible ideas. Changing the physical properties of the universe has no bearing whatsoever about whether 163 is prime or whether there exists a set that contains all other sets. – Greg Martin Feb 18 '19 at 19:54
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    Science fiction author Greg Egan has had such thoughts; if I recall correctly, his collection Dark Integers and Other Stories handles this topic in the title story and at least one other. – FredH Feb 18 '19 at 21:21
  • There is journal Philosophia Mathematica by Oxford University press and arxiv.org contains at least 50 articles about hypercomputation. That gives me some strength while watching how my question is downvoted. – TomR Feb 18 '19 at 21:45
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    @TomR Hypercomputation is indeed a well-researched area - but I don't see what that has to do with your question (hypercomputation certainly doesn't require any new logical system). Your question is being downvoted (at least by me) because it is simultaneously too unclear and too broad. A question which may help focus things: what do you see being part of a successful answer to this question? – Noah Schweber Feb 19 '19 at 15:25
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    Also, how do black holes specifically enter into this? – Noah Schweber Feb 19 '19 at 15:26
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    @Noah Schweber: It was obvious to me that black holes are neither here nor there regarding the philosophical issues, at least that we know of (for all we know, $F=ma$ or entropy might be seen as relevant by our dark matter descendants in a few million years). However, not knowing what "hypercomputation" is, I wasn't sure and didn't bother looking it up, thinking maybe it might be sufficiently "out there" (like relevance logic, maybe). Motivated by your comment, I looked it up --- it's just a new name for the old notion of using oracles. – Dave L. Renfro Feb 20 '19 at 09:48
  • Nothing is more powerful than oracles and as Goedel/Church/Turing prohibits the oracles in this world, hence we need New Physics for getting oracles. – TomR Feb 20 '19 at 09:51
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    @Noah Schweber: I wonder why the Wikipedia article on turing degrees is not cited in the Wikipedia article on hypercomputation. – Dave L. Renfro Feb 20 '19 at 09:56
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    "we need New Physics for getting oracles" --- I'm afraid invoking black holes and quantum mechanics and such will seem as quaint in several hundred years as those who made similar statements around 1800 about magnetism or those who made similar statements around 1900 about radioactivity. If you spend some time pursuing old scientific literature and speculations (almost all now digitized and freely available on the internet), you'll find that pretty much each era's "cutting edge science" was used in philosophical speculations in a way that now seems "not even wrong". – Dave L. Renfro Feb 20 '19 at 10:03

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Logic by itself does not apply to our universe, you also need physics. For example, if you had two sheep and four sheep and wanted to know how many sheep you had in all, you would have to use physics to map sheep to integers, logic to deduce $2+4=6$, and then physics again to map $6$ back to sheep. The idea that physical objects can be mapped to integers for counting is perhaps the first physical law that people learn.

In this light, it is not exactly clear what it would mean for an alternate universe to "not follow logic." If in the alternate universe two and two made five, you could get that to work with the same math we have now: all you would have to do is have a different physical law of counting, that said that sheep in a field should be mapped on to something other than integers, and sheep composition should be something other than addition. If nothing in our present knowledge of math would suffice, you could just invent something new.

Are there any systems that can be conceived of that cannot be constructed at all? Sure, one of them is a list of instructions that would allow you to, by following them, determine whether or not a computer program will eventually stop running. However, if we lived in a universe where the laws of physics "solved that problem," we could still make the non-constructive observation that doing such-and-such a thing will cause the universe to deliver us the answer to the Halting problem. Doing so would pose no threat to logic, because that would just mean the universe was at least one step above a Turning machine on the hierarchy.

Granted, the fact that I can't think of any situation that would force us to abandon logic is not proof of the claim that there is no such situation. Still, the constructive power at hand is great - perhaps great enough to handle any situation we could conceive of.

Display Name
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  • Counting is not a physical law. You don't have to "map sheep to integers" to count them, and counting them certainly doesn't require physics in any way, shape, or form. – Matt Samuel Feb 18 '19 at 20:57
  • How can you count without mapping things to integers? You point at the first and say, "one," the next and say, "two," and so on. That's clearly a mapping. Secondly, you will not be able to count unless you believe that a physical quantity (the number of sheep in your pen) corresponds to a mathematical model (the number in your head.) If that's not physics what is? ;) – Display Name Feb 18 '19 at 22:15
  • Some animals can count. Are they mapping the objects to integers? Are they doing physics? – Matt Samuel Feb 18 '19 at 22:21
  • I'm not sure if you could tell whether or not an animal was thinking about ones, twos, and threes when "counting." However people that learn to count have a concept of numbers for sure. Children who are past finger counting are associating the things they count with conceptual abstract numbers. – Display Name Feb 18 '19 at 22:39
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    @Matt Samuel: Counting is not a physical law. --- In my opinion the phrase "physical law" appears to set up a straw man by being too specific of a term for the kind of philosophical considerations involved, because I suspect Display Name is using the term "physics" in a more general sense than you are. Regarding "doesn't require physics in any way, shape, or form", many highly thoughtful people would disagree when "physics" is interpreted in a more general philosophical sense. Did you read my cited (in another comment here) Stack Exchange answer? – Dave L. Renfro Feb 19 '19 at 10:41