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The theory would explain why the laws of physics tend to break down in some instances. I prefer to believe there is so much we don't understand about the universe.
I think I know what mathematical analysis is: infinite series, differential calculus, integral calculus, plus probably other things. Feel free to complete the list if you think I am missing important areas.Literally the entire point of calculus is to look at the behavior of the series as it approaches an infinite length, not to actually calculate the series out. You don't need some mystical perception/apprehension to make those calculations. Humans don't have some mystical ability to understand how a series converges or what it converges to.
Mathematical analysis is a fundamentally different topic than calculus. The reason computers lag behind humans at the highs of the field are likely similar to the reasons computers lagged behind humans at Chess for so long and still do at Go. It's not simply about applying strict rules, but about recognizing the abstract patterns. That has nothing to do with infinity and everything to do with creativity, but that's an entirely different topic and has nothing to do with calculus.
I think I know what mathematical analysis is: infinite series, differential calculus, integral calculus, plus probably other things. Feel free to complete the list if you think I am missing important areas.
The idea of infinity implies the idea of direction.
Regarding (a): I think continuity, and thus the infinitesimal, is implicit to ordinary experience / perception.
This might be another one: how is any program supposed to decide if a given collection of numbers is the result of a function? Suppose it is.
How could it decide if that sequence is an infinite series? Suppose it is.
How could it decide that that series is countable or uncountable?
Thank you for sharing. It reminds me of Dedekind's paper "Continuity and irrational numbers". Cuts are the elements on the real number line and needed to be posited for continuity on account of the gaps by the rational numbers seen as constituting the abstract number line, in order to realize a full line.Mathematical analysis is the study of continual change. Yes, there's calculus involved, calculus is really the underpinning of it. If you want to look at it this way, mathematical analysis is the theory, calculus is the language. Computers are absolutely capable of performing calculus, just head over to Wolfram Alpha if you want a demonstration.
The idea of infinity implies the idea of direction in the sense of mathematical series. The idea of the infinitesimal to realize continuity, is given in relation to its forming a set of numbers greater and a set of numbers less than. This implies directionality.Why?
I think continuity, and thus the infinitesimal, is implicit to ordinary experience / perception because we if we don't think of two discrete time events, say marked by a sort of self-consciousness in the present at two different moments, but rather a being in the zone or awareness of duration, then that is the sort of flow to time that implies a sort of continuity not wholly unlike that of Dedekind's cuts. Or, even better, one can take a moment of self-consciousness as being akin to a cut.Can you explain what you mean by this?
Fair enough. A lot of what you're saying sounds either constructivist or finitist.It is impossible to tell, given a set of numbers and no explanation as to how they appeared, if they were the result of a single function, multiple functions, or fully randomly generated (assuming true randomness even exists). That holds true for humans as well as computers.
It is possible (again for both humans and computers) to develop a potential algorithm which would generate said collection of numbers, but you cannot say if the numbers were or were not generated by that algorithm.
Again, without a description of how the numbers were generated, it would be impossible for both humans and computers to tell if a sequence is infinite or simply larger than has been currently experienced.
Re: your suggestion that we look for a one-one correspondence and see if we get it or not, is that the set of reals between any two numbers on the real number line, as well as naturals and rational numbers I suppose, is that ellipses (". . .") needs to be conceptually / relationally filled-in so to speak by our abstract thought / attunement as individuals. This requires a sort of self-consciousness, for instance, seeing how the context of a proof is formulated as being something we are looking to demonstrate in thought. Otherwise there is no way of knowing that uncontability has been demonstrated for the real numbers. There is a sort of intentionality to the act that is simply not present in Turing Machines / computers. So, I think you were correct in suspecting that self-consciousness or the ability to think is a crucial difference between calculus for us and for computers, like you said earlier in our discussion.By finding a 1:1 mapping between the natural numbers and the given set, or by proving that no such mapping exists. Again, computers would do this the same way humans would.
The fault in your reasoning is assuming that a problem we have with our computers and our math is a problem that those who run the simulation would have. We really have no way of knowing what the capacities of the system running the simulation are unless those who created it confirm its existence by informing us of it. Like I speculated earlier, the rules that govern our existence could be a stripped down and simplified version of the rules that govern theirs.
The computer system running this damned thing could somehow be analog for all we know. It could also be a 4 dimensional object. We really have no way of knowing unless they tell us OR we come across a fault in reality that we can use to gain access ourselves. Of course by coming across this fault, it's quite possible we may just corrupt our own existence to the point of it being unrecoverable and thus be our own doom, but thems the breaks when you cause the biggest bluescreen of all time.
We should probably make sure we trigger the equivalent of File->Save first. Otherwise some weird alien blob could lose 14 billion years of work.
Fair enough. A lot of what you're saying sounds either constructivist or finitist.
Re: your suggestion that we look for a one-one correspondence and see if we get it or not, is that the set of reals between any two numbers on the real number line, as well as naturals and rational numbers I suppose, is that ellipses (". . .") needs to be conceptually / relationally filled-in so to speak by our abstract thought / attunement as individuals. This requires a sort of self-consciousness, for instance, seeing how the context of a proof is formulated as being something we are looking to demonstrate in thought. Otherwise there is no way of knowing that uncontability has been demonstrated for the real numbers. There is a sort of intentionality to the act that is simply not present in Turing Machines / computers. So, I think you were correct in suspecting that self-consciousness or the ability to think is a crucial difference between calculus for us and for computers, like you said earlier in our discussion.
On the possibility of a Planck scale interpretation, fair enough. Perhaps my interpretation is a bit Kantian in supposing an abstract category to be informing that sort of ordinary experience.I suppose I make a break between conceptual arguments and practical arguments. Conceptually, the infinite and the infinitesimal are completely valid. As far as our physical universe, I'm not sure I'd say they don't as much as I don't think it matters.
I don't think true continuity is needed for how we experience the universe. Planck length and Planck time are so much smaller than anything we can experience or measure that I'm not at all convinced a system based on them as a foundation would have any discernible difference to us than a system with true continuity.
Note: I'm not arguing our universe is based on Planck scale, I'm arguing that if it was we wouldn't be able to tell the difference and thus any arguments about infinite/infinitesimal being required to simulate it don't hold water for me.
Cantor's diagonal argument doesn't use infinity, or the concept of infinity, directly. It uses the concept of a finite, but arbitrarily large, list and proof by contradiction.
Honestly, what does it matter?
To be a 100% fail safe simulation, there can be absolutely no bugs, errors or mishaps - or the subjects would realize that it in fact IS a simulation.
Even if there were a computer or an intelligence powerful enough to pull this off, I find it unlikely that it has run for hundreds of billions of years without the slightest hick-up.
Or is the idea that distant history never really took place, but is just memories baked into the simulation?
Dinosaurs never really existed, the simulation simply added bones buried across the globe, to make us think they actually existed at one point?
If I ran a simulation and realized that something didn't fully work or was believable, I would simply shut down, improve and reboot.
Reboot = Big Bang?
Or you simply program the subjects to view any bugs in a certain way. Provide us with science, for instance, to explain everything that we're unsure of.
Even more rational, bake religion ibto the equation to make people never question anything at all.
To be a 100% fail safe simulation, there can be absolutely no bugs, errors or mishaps - or the subjects would realize that it in fact IS a simulation.
Or you simply program the subjects to view any bugs in a certain way. Provide us with science, for instance, to explain everything that we're unsure of.
Being skeptical about the entire universe in its entirety being a simulation gets at why one thinks existence is true / objects are real. I think this is a valuable question for its own sake.Honestly, what does it matter?
To be a 100% fail safe simulation, there can be absolutely no bugs, errors or mishaps - or the subjects would realize that it in fact IS a simulation.
Even if there were a computer or an intelligence powerful enough to pull this off, I find it unlikely that it has run for hundreds of billions of years without the slightest hick-up.
Or is the idea that distant history never really took place, but is just memories baked into the simulation?
Dinosaurs never really existed, the simulation simply added bones buried across the globe, to make us think they actually existed at one point?
If I ran a simulation and realized that something didn't fully work or was believable, I would simply shut down, improve and reboot.
Reboot = Big Bang?