Comments by "" (@tinkeringtim7999) on "Insights into Mathematics"
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Mathematics is, I think you'll agree, the map and not the territory. You may not agree but must eventually, that the only working maps can be those which assume a continuous territory. And I mean that in the topological sense, rationals are a continuous space.
Do you agree that an affine space is continuous? If so, how do you justify your other comment that the universe may be signalling it is discreete? Can you see how it would be impossible to do the same with just an integer lattice - not using the notion ofnthe space between lattice points since distance again as a concept inherently includes continuity in the existence of positions from one position to another - if not, it could not move past the discontinuity.
If we don't have underlying continuity, we also don't have causal relationships, if we did, then on that space it would be continuously connected again.
Reality is fundamentally continuous, discreeteness has only ever been found in abstraction layers we've added onto continuous reality in order to make a finite picture in our brains.
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@njwildberger Well, what it means is is, if the quantity computed by the computer is called a number, then if it continuously passes from 1 unit to 2 units, the whatever (continuously existing) physical state it is in is a computational representation of that "number". It doesn't blip out of existence everytime it encounters what our unit scale happens to regard as an irrational point.
If for you, numbers are by their very definition combinatorial, then you could not be satisfied really with any analogue computer, although they work.
If numbers are not actually fundamentally combinatorial, but those are merely a class of numbers, then it's perfectly reasonable to use any symbol & operation whose algebra is the same as that of a field, as a number.
Bottom line is, your computability criterion strictly applies to "universal" computers, which are ironically a subset of possible computers.
Every physical process is a computation, follows from Landau's principle.
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@20-sideddice13 That's an article of faith, and proves the entire debate has gone over your head.
Recap, the debate is constructionist vs formalist. Formalists won control of institutions for a number of reasons, none of them logical. Now generations only know the formalise view point. Manufactured "consensus" by hiding the debate for generations.
All you're proving is a PhD might be well educated to be a cog in that machine, but is utterly ill-equipped to tackle questions about truth and the fundamentals of mathematics. It's seems clear that your PhD has in fact been a handicap for this kind of question.
The who bunch of garbage about the 5th postulate as if nobody thought of curved surfaces for a thousand years is just historically illiterate garbage which is pumped out as truth today, largely because it made Hilberts career with T axioms.
Tell me, is the Banach-Tarsky "Paradox" a "Paradox" or a proof by contradiction? When a formalist declares "I am unable to conjure a false dichotomy, if I propose a dichotomy, it's as valid as any other" - anyone else saying that would be obviously insane. But put fancy symbols round it and use political power to crush opponents careers, and Hilbert easily changed "Theology" to "The future of mathematics". Every win was by dirty politics using extraordinary clout and his cabal at Göttingen. Literally nothing was won by strength of logic. The opposite in fact. His whole program was so defeated he had to leech off physics to try to get some credibility. Yet another confidence trick worked.
Of course, the Victors wrote the history. I could go on, quaternions are another tragic example.
You have a lot to learn, having a PhD puts you on the first rung. Good luck.
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