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> they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself

> they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself part of reality, i.e. does not exist in the most obvious sense of the word. They are not arbitrary products of the mind, for the reasons you explain; but they are products of the mind. The property of NP-completeness didn't exist until we invented algorithms and analysed their time complexity, though many problems do posses that property; and in the same way the property of triangularity didn't exist until Greeks started imagining geometry in terms of idealized regions of space, defined in terms of simple relations between points without size. (And then, the triangularity property changed a lot when people started questioning the parallel postulate and discovered non-euclidean geometry). It is not necessary that an entity exists for it to form impressions in the mind, it is enough that the mind can imagine it from actual existing perceptual elements, and our imagination fills in the details. Is it truly needed for a monster to exist under the bed for it to impress our juvenile minds?

Koshkin 4y ago
> they are products of the mind Well, a fighter jet, too, is a product of the mind. > triangularity didn't exist until Tell this to triangular molecules. > a monster This says more about people than it does about the reality (though even fears and phantasms can indeed be rooted in reality).
- TuringTest 4y ago
  > Well, a fighter jet, too, is a product of the mind. I'd say there's a difference between having a physical, tangible fighter jet in front of you, with all the connections between its molecules in a configuration that allows it to fly, and having "the property of being a fighter jet" in front of you. > Tell this to triangular molecules. Do they have "a triangular" in them? https://www.smbc-comics.com/comic/real-2 https://www.smbc-comics.com/comic/real-2 > This says more about people than it does about the reality Right, and having people believe that properties exist as an entity outside of the mind says more about people than it does about the reality.
archi42 4y ago
> The property of NP-completeness didn't exist until we invented algorithms and analysed their time complexity [...] Now that is a bolt statement to make with such certainty :) Imagine it's 1822 and you're a salesmen, travelling from town to town to sell your wares. Of course you want to save on time and distance travelled, so you'd like to pick the shortes route that covers all towns on your list. Now how complicated can that be? I'd dare to say the probem you're facing was already NP-complete back then, you just didn't know if you're too stupid (no, you're not!) or if it was in fact impossible without trying every possible route (yes). In other words: When inventing a new predicate (e.g. is_np_complete) in non-temporal, binary logic, that predicate is always true or false (or undecided :)) for a given input. You essentially say that the predicate only exists once it has been created, but I say it has merely been given a name to reference it. I'd like to present a logical argument why I am right beyond a doubt, but this might be undecidable; at least I'm stuck thinking about it, much like the imaginative you from 1822 (my problem is that I don't know if it's possible to enumerate all possible predicates [that map each possible input to every possible output {true, false, undecidable}] - if it is I think I can make a sound argument). And very much like that fictive person should probably just start travelling on a good enough route, I should also get back to work ;) Maybe you can come up with a formal proof why a predicate can only exist once it has been formalized for the first time? //edit: some clarification to show that this is a friendly discussion :)
- TuringTest 4y ago
  > some clarification to show that this is a friendly discussion :) Thanks, people assume the worst intentions because of the lack of tone in written messages. :-) > You essentially say that the predicate only exists once it has been created, but I say it has merely been given a name to reference it. I don't see how this distinction makes much of a difference. Does a predicate exist if no human is thinking about it? Before anyone defined the problem for the first time, the problem didn't exist and therefore it couldn't have properties. Unless I was salesperson with a singularly mathematical mind, I would not try to solve the problem for every possible case but just for the particular set of cities that I traveled through. And if I happened to be a rural Chinese, [1] ;-) I may well be able to solve my problem in polynomial time. NP-completeness is a property of a family of problem instances, so it matters only when you are studying the whole family, even though it may affect any particular instance (or not). [1] i.e., there may be subsets of the general problem that may be solved in polynomial time, and my particular instance may belong to that subset. See Rural postman problem and Chinese postman problem https://kups.ub.uni-koeln.de/54671/1/rep-92.113-koeln.pdf https://kups.ub.uni-koeln.de/54671/1/rep-92.113-koeln.pdf https://en.wikipedia.org/wiki/Chinese_postman_problem https://en.wikipedia.org/wiki/Chinese_postman_problem However, let's assume that we have defined the problem in its general terms. So, does the property of being a hard problem exist as soon as you state it, as an inescapable consequence of its formal definition, even before someone starts to study its complexity? I'd say absolutely yes, and if that's what you mean saying that the property 'exists', then we are on the same page. > Maybe you can come up with a formal proof why a predicate can only exist once it has been formalized for the first time? Yest, I think I could do that if I tried. I also think that I could do the opposite if I tried, showing that any predicate exists from the start of time, just waiting to be discovered. You see, the problem with formalism is that the theorems that can be proven depend completely on the assumptions you incorporate when defining a specific formal system; therefore, I can orient the reasoning towards one conclusion or the other as I am interested, as long as it does not incorporate a set of axioms and rules of inference that produce an internal contradiction. Formal systems are most valuable because they allow us to get rid of inconsistent assumptions in our reasoning, not necessarily because those statements correspond exactly one-to-one to entities in the real world.

Koshkin 4y ago

> they are products of the mind Well, a fighter jet, too, is a product of the mind. > triangularity didn't exist until Tell this to triangular molecules. > a monster This says more about people than it does about the reality (though even fears and phantasms can indeed be rooted in reality).

TuringTest 4y ago
> Well, a fighter jet, too, is a product of the mind. I'd say there's a difference between having a physical, tangible fighter jet in front of you, with all the connections between its molecules in a configuration that allows it to fly, and having "the property of being a fighter jet" in front of you. > Tell this to triangular molecules. Do they have "a triangular" in them? https://www.smbc-comics.com/comic/real-2 https://www.smbc-comics.com/comic/real-2 > This says more about people than it does about the reality Right, and having people believe that properties exist as an entity outside of the mind says more about people than it does about the reality.

archi42 4y ago

> The property of NP-completeness didn't exist until we invented algorithms and analysed their time complexity [...] Now that is a bolt statement to make with such certainty :) Imagine it's 1822 and you're a salesmen, travelling from town to town to sell your wares. Of course you want to save on time and distance travelled, so you'd like to pick the shortes route that covers all towns on your list. Now how complicated can that be? I'd dare to say the probem you're facing was already NP-complete back then, you just didn't know if you're too stupid (no, you're not!) or if it was in fact impossible without trying every possible route (yes). In other words: When inventing a new predicate (e.g. is_np_complete) in non-temporal, binary logic, that predicate is always true or false (or undecided :)) for a given input. You essentially say that the predicate only exists once it has been created, but I say it has merely been given a name to reference it. I'd like to present a logical argument why I am right beyond a doubt, but this might be undecidable; at least I'm stuck thinking about it, much like the imaginative you from 1822 (my problem is that I don't know if it's possible to enumerate all possible predicates [that map each possible input to every possible output {true, false, undecidable}] - if it is I think I can make a sound argument). And very much like that fictive person should probably just start travelling on a good enough route, I should also get back to work ;) Maybe you can come up with a formal proof why a predicate can only exist once it has been formalized for the first time? //edit: some clarification to show that this is a friendly discussion :)

TuringTest 4y ago
> some clarification to show that this is a friendly discussion :) Thanks, people assume the worst intentions because of the lack of tone in written messages. :-) > You essentially say that the predicate only exists once it has been created, but I say it has merely been given a name to reference it. I don't see how this distinction makes much of a difference. Does a predicate exist if no human is thinking about it? Before anyone defined the problem for the first time, the problem didn't exist and therefore it couldn't have properties. Unless I was salesperson with a singularly mathematical mind, I would not try to solve the problem for every possible case but just for the particular set of cities that I traveled through. And if I happened to be a rural Chinese, [1] ;-) I may well be able to solve my problem in polynomial time. NP-completeness is a property of a family of problem instances, so it matters only when you are studying the whole family, even though it may affect any particular instance (or not). [1] i.e., there may be subsets of the general problem that may be solved in polynomial time, and my particular instance may belong to that subset. See Rural postman problem and Chinese postman problem https://kups.ub.uni-koeln.de/54671/1/rep-92.113-koeln.pdf https://kups.ub.uni-koeln.de/54671/1/rep-92.113-koeln.pdf https://en.wikipedia.org/wiki/Chinese_postman_problem https://en.wikipedia.org/wiki/Chinese_postman_problem However, let's assume that we have defined the problem in its general terms. So, does the property of being a hard problem exist as soon as you state it, as an inescapable consequence of its formal definition, even before someone starts to study its complexity? I'd say absolutely yes, and if that's what you mean saying that the property 'exists', then we are on the same page. > Maybe you can come up with a formal proof why a predicate can only exist once it has been formalized for the first time? Yest, I think I could do that if I tried. I also think that I could do the opposite if I tried, showing that any predicate exists from the start of time, just waiting to be discovered. You see, the problem with formalism is that the theorems that can be proven depend completely on the assumptions you incorporate when defining a specific formal system; therefore, I can orient the reasoning towards one conclusion or the other as I am interested, as long as it does not incorporate a set of axioms and rules of inference that produce an internal contradiction. Formal systems are most valuable because they allow us to get rid of inconsistent assumptions in our reasoning, not necessarily because those statements correspond exactly one-to-one to entities in the real world.