i — Mephist
you cannot count on the fact that you always get the same output for the same input with absolute certainty: you get results that are statistically determined, but not deterministic. — Mephist
Constructive physics (constructivist logic) can ASSUME the existence of a function that you can call "random" (whatever it means: it's an axiomatic theory), representing a physical process. Only that you cannot DERIVE or COMPUTE this function. You have to assume it as an axiom of the theory. The point is that this is allowed by the logic because you cannot introduce inconsistencies in this way! — Mephist
You cannot test if it is countable or not with physical experiments limited in time (how do you know that you don't get the same results again after a — Mephist
One way to define "constructive physics" is simply to say, "it uses constructive mathematics". But definitions of the latter sometimes arise principally from avoiding the LEM. Another tack is to avoid non-computable numbers. Or simply to state that experiments must be conclusive in a reasonable finite amount of time. I'm not sure what you two are referring to here. But I haven't read all the thread. — jgill
How do you measure how much we need to know about something before we can name it? — fishfry
On the contrary, the day they discovered that the galaxies are spinning too fast to hold together, they named the cause "dark matter" while having no idea what it is or whether it exists at all. — fishfry
That explains a lot. Why should I (or anyone else) accept the constraints of your peculiar language?I cannot communicate with someone who doesn't speak my language. — Metaphysician Undercover
How could you ever make such a determination, given your admission that you are unwilling even to try to understand my (or others') usage of the terms, simply because it is different from yours?It strikes me that you have disregard for the fundamental rules of logic ... — Metaphysician Undercover
But instead of facing this fact, that the theories are wrong, someone has dreamed up a name "dark matter", and they attribute the fact that the theories are wrong to this mysterious thing, "dark matter". Why not just call it like it is, "the theories are wrong", dump the theories, and the "dark matter" which the theories necessitate because they're wrong, and get on with producing a new theory which doesn't make this mistake? — Metaphysician Undercover
That explains a lot. Why should I (or anyone else) accept the constraints of your peculiar language? — aletheist
How could you ever make such a determination, given your admission that you are unwilling even to try to understand my (or others') usage of the terms, simply because it is different from yours? — aletheist
If you cannot see that the truth of this statement indicates that the theories have been falsified, then I'm afraid your denial is beyond hope.By all our known theories of physics, galaxies should have flown apart long ago.
If you cannot see that the truth of this statement indicates that the theories have been falsified, then I'm afraid your denial is beyond hope. — Metaphysician Undercover
Will someday be falsified', is not the same as 'has been falsified'. The falsification is what determines the faults, demonstrating the weaknesses of the theory, showing us where improvement is needed. There is no point in dismissing theories which have not yet been falsified, because we would not know what needs to be improved. That's the scientific method, observations which are inconsistent with what the theory predicts reveal the faults in the theory. But until those inconsistent observations come about, we don't know where the weaknesses of the theory lie. — Metaphysician Undercover
None of those dictate the peculiar metaphysical definitions that you insist on imposing for terms like "existence" and "object," even in the context of non-platonist mathematics where they entail nothing ontological whatsoever.The constraints of my language are the fundamental laws of logic, identity, non-contradiction, excluded middle. — Metaphysician Undercover
On this we agree.... I see no point in continuing. — Metaphysician Undercover
But you asked, "What if there exists no such thing in nature?" But that is not a problem for math, because math is not physics. In math there is certainty about the repeatability of functions. — fishfry
Why are you trying to convince me that math isn't physics? I'm talking about the measure of computable bitstrings in the space of all bitstrings. The measure of the computable bitstrings is zero. — fishfry
How do you define this measure in pure mathematical terms? You cannot use probability, because probability is physics (unless you find a sound mathematical definition of probability) — Mephist
. . . but mathematics need computations for proofs. — Mephist
If a computation is too long to be performed by any computer even in principle, is it still valid? — Mephist
I think that you are ascribing to mathematics the kind of role that I don't think it has. At least, not directly. Or maybe I did walk into this when elaborating over my example. Its aim wasn't to model the structure of physical objects, but to illustrate how coarse structures not literally represented by mathematical ideals, can still be usefully approximated by those ideals. It was designed to have some similarity with the atomic structure of materials. But it is not a theoretical model for physics. The idea was, that actual physical structures approximate the mathematical ideals, and our numerical algorithms approximate those same ideals, and thus, under certain assumptions of the magnitudes of the involved deviations, our numerical algorithms match the physical structures within the required precision.Now we have entered into an extremely confused and contradictory conception within which distinct things are said to be distinct particulars, and they are treated by the application of the theory as distinct particulars, yet they are stipulated by the assumptions of that same theory to be the same in an absolute way. That's the kind of mess which "grain uniformity" might give us. — Metaphysician Undercover
Which theory do you mean? For me, real numbers theorize some characteristics of computation. And do so imperfectly. The algebraic structure is defined over some converging computational sequences. It does allow for imaginary objects that do not correspond to actual computational processes, because the latter can not be specified procedurally. And thus it works with incomplete specification. But it is a best effort theory. I am not necessarily subscribed to the idea that real numbers correspond to the points of physical lines, whatever that may mean. It may turn out that this conceptualization works, but I am not convinced. I do think that it works for approximations however.But approximation in practise is not the same as approximation in theory. — Metaphysician Undercover
But what I am saying is that you can equally well assume as an axiom (that would be incompatible with ZFC) that Turing machines DO NOT exist! — Mephist
Constructivist theories correspond to elegant constructions in topology, represented as internal languages of certain categories. In comparison, ZFC axioms seem to be much more arbitrary, from my point of view. — Mephist
I've never heard of this idea, that TM's don't exist. I see no problem expressing TMs in set theory. An unbounded tape of cells is modeled as the integers. You have some rules that let you move right or left. It's pretty straightforward.
Can you say more about this? I have never heard this idea at all. It seems VERY restrictive. Perhaps it's like denying the axiom of infinity. Logically consistent but too restrictive to do math with. — fishfry
I don't know what would be the equivalent limitation to Turing machines that corresponds to dependently typed lambda calculus (if there is one). So, I should have said that we can assume that the "original" (non limited) Turing machine does not exist — Mephist
P.S. Here's a citation taken from wikipedia: — Mephist
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