Can you picture a hypersphere as easily as a sphere? Does that make you doubt that it is a constructable object? Is your whole argument going to be based on what you personally find concretely visible in your minds eye? That’s a weak epistemology that won’t get you far. — apokrisis
I can’t picture a cut which doesn’t result in a gap. — apokrisis
What exactly do you mean by this? I don't think 'a state of everything' needs to exist for 'something' to exist.
— keystone
Can you picture getting something from nothing? Can you picture being left with something having carved away most of everything?
One of these two is more picturable, no? — apokrisis
You're the first to ever entertain my idea on cutting a continuum. (or perhaps you have the same idea)
— keystone
It’s a standard kind of idea. For instance - https://en.wikipedia.org/wiki/Dedekind_cut — apokrisis
You return to the point that 'each is the measure of the other' so I think that's key to your argument, I'm just not comprehending it yet...
— keystone
It’s the logic of a reciprocal or inverse operation. — apokrisis
Hilbert's Hotel is an imaginary analogy that seems fine to me. — TonesInDeepFreeze
Yes, those are paradoxes. But my point is that they are not contradictions in ZFC* (and I'm not claiming that you claimed that they are contradictions in ZFC). — TonesInDeepFreeze
Zeno's paradox is actually resolved thanks to ZFC (I mean thanks to ZFC for providing a rigorous axiomatization for late 19th century analysis). — TonesInDeepFreeze
Galileo's paradox strikes me a "nothing burger". I am not disquieted that there is a 1-1 between the squares and the naturals. — TonesInDeepFreeze
Dartboard paradox. I don't know enough about it. — TonesInDeepFreeze
Thompson's lamp. A non-converging sequence, if I recall. Again, rather than this being a problem for set theory, it's a problem that set theory (as an axiomatization of analysis) avoids. — TonesInDeepFreeze
A contradiction in ZFC would be a theorem of the form:
P & ~P
No such theorem has been shown in ZFC. — TonesInDeepFreeze
can you imagine the actual endless hotel as a whole? — keystone
whether you can imagine a set of all natural numbers. — keystone
My point is that your description is not an accurate or even reasonable simplification of how set theory proves that there is a complete ordered field and a total ordering of its carrier set. (The carrier set is the set of real numbers and the total ordering is the standard less-than relation on the set of real numbers.) — TonesInDeepFreeze
I feel like you could give me a little more slack here on my phrasing.
— keystone
Your phrasing struck me as polemical and misleading by saying "magic" and "leap", which does not do justice to the fact that set theory is axiomatic, and while the set of naturals is given by axiomatic "fiat", the development of the integers, rationals and reals is done from the set of naturals in a rigorous construction. — TonesInDeepFreeze
What was the 19th century analysis resolution to Zeno's paradox? — keystone
Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms? — keystone
I don't think calculus needs actual infinity to work. — keystone
I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either. — keystone
I have no problem forming a 1-1 relationship between n and the n^2. I just don't think there's an actual set that contains all n (and similarly all n^2). In other words, my qualms are not with the math, they are with the philosophy. — keystone
consider an alternate interpretation: in the first step 1 there is a dislodged guest, in the second step there is a dislodged guest, in the third step there is a dislodged guest [etc.] — keystone
My point is that your description is not an accurate or even reasonable simplification of how set theory proves that there is a complete ordered field and a total ordering of its carrier set. (The carrier set is the set of real numbers and the total ordering is the standard less-than relation on the set of real numbers.)
— TonesInDeepFreeze
I'm not sure what you're referring to. — keystone
The Diagonal Paradox can be extended in principle to any curve in 2D. For example, a circle of radius 1 has a circumference of 2pi, but if I apply my system of sine curves to the circumference I find that as they converge uniformly to the existing circumference, their lengths tend to infinity. Hence I am staring at what appears to be the simple circle I began with, but I now have one with infinite circumference, and hence infinite area.
Thus infinity is everywhere in plane geometry where it shouldn't be. — jgill
I consider the Paradox an aberration that results from collapsing one dimension to a lower dimension in certain circumstances and insignificant although bizarre. But Wolfram claims that this crops up in Feynman diagrams. It goes to the very nature of lines and points. — jgill
I consider the Paradox an aberration that results from collapsing one dimension to a lower dimension in certain circumstances and insignificant although bizarre. But Wolfram claims that this crops up in Feynman diagrams. It goes to the very nature of lines and points. — jgill
No, I think it is significant and general. — apokrisis
I believe that irrationals are algorithms which describe this mysterious other object - continua. — keystone
I don't believe there is a fundamental length since any length can be divided. — keystone
I appreciate that you are using a lot of physics analogies here but I feel like you've gone to far. — keystone
I can imagine a mind that lives in a 4D universe that can picture a 4D hypersphere as easily as a sphere. — keystone
In this analogy, the interaction is the act of cutting. — keystone
If all potential images popped out simultaneously then the whole page would pop out resulting in no image at all. — keystone
The page contains the potential of infinite images, — keystone
Can the image and background be the 'measure of the other' that you're referring to? If so, then that makes sense to me. — keystone
Are you not disquieted that a probability of 0 does not mean impossible? — keystone
Of course, non-infinitistic systematizations for mathematics are interesting and of real mathematical and philosophical import. And there are many systems that have been developed. Personally though, I am also interested in comparisons not just on the basis of having achieved the thing, but also in how complicated the systems are to work with, the aesthetics, and whether fulfilling the philosophical motivations are worth the costs in complication and aesthetics. — TonesInDeepFreeze
I think I see now. You didn't mean that Cantor claims that we can list the points in the line, but rather Cantor showed that we can't do that?
If you let me know that the above is correct, then I should retract what I said earlier. — TonesInDeepFreeze
If the infinitistic systemization for mathematics are more powerful, beautiful, and simple — keystone
I do wonder whether our infinitistic systemization can simply be reinterpreted from being based on actual infinity to being based on potential infinity. — keystone
So, at least as far as I can tell, saying 'potentially infinite' is not yet, at least, a formalized notion but rather a manner of speaking — TonesInDeepFreeze
I'm thinking of something more irreducibly complex. A dimensionality that is "completely" void can't help but have some residual degree of local fluctuation. And likewise, a dimensionality that is "completely" full, can't help but have some residual degree of fluctuation – but of the opposite kind. — apokrisis
However what really matters – if we are interested in models of reality as it actually is – is the fact that finitude can be extracted from pure unboundedness. — apokrisis
Although there are still big questionmarks. We still seem to need eternal inflation at the front end as a kind of somethingness to get the Big Bang ball rolling — apokrisis
We do prove "sqrt(2) is a [real] number". — TonesInDeepFreeze
In QM we have come to accept a certain level of uncertainty. Why can't we do the same in math?
— keystone
I wouldn't argue that we can't. I suppose people already have made logic systems with values such as 'uncertain' that can be be applied to a different mathematics. And I can imagine that certain scientific enquires might be better served by such systems.
But that doesn't erase the rewards meanwhile of classical mathematics. — TonesInDeepFreeze
However, I feel like you're mentioning a lot of physical phenomenon but not explaining clearly how they relate to mathematics. — keystone
I can draw a line with open ends on a piece of paper and label the ends negative and positive infinity. This unbounded object is entirely finite. — keystone
It appears that you are looking at the universe from a point-based perspective in that there's a first instant which is followed by the next instant, and so on. — keystone
My impression is that we do not prove sqrt(2) is a number, but instead we assume it is a number by means of the completeness axiom. — keystone
So time has a fundamental grain determined by c. A moment or duree is the completion of a change. And the Planck scale is the size of the smallest such moment. — apokrisis
Definition of the second
In 1968, the duration of the second was defined to be 9192631770 vibrations of the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom. Prior to that it was defined by there being 31556925.9747 seconds in the tropical year 1900.[18] The 1968 definition was updated in 2019 to reflect the new definitions of the ampere, kelvin, kilogram, and mole decided upon at the 2019 redefinition of the International System of Units. Timekeeping researchers are currently working on developing an even more stable atomic reference for the second, with a plan to find a more precise definition of the second as atomic clocks improve based on optical clocks or the Rydberg constant around 2030.
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.