That differs from how I find 'classical' is used. I find that 'classical' mathematics means all and only those results that can be formalized as theorems of ZFC with classical logic. And classical logic means the first order predicate calculus including the law of excluded middle. — TonesInDeepFreeze
In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory.
I don't see any advantage to the fact that your way of conceptualizing pi is "entirely finite." — T Clark
It is my understanding that computers do not generally store the algorithm for generating pi, they store the actual number rounded to a specified number of decimal laces. If computers think pi is a number, why shouldn't I?" — T Clark
When I measure light one way, it's always a wave. When I measure it another way, it's always a particle. It's not a wave that becomes a particle. It's always both at the same time." — T Clark
The universe has a wonderful way of avoiding actual infinities.
— keystone
Again, sez you — T Clark
I think you and I have taken this as far as we're going to get. I don't see the need for or value of the way of seeing things you propose. You obviously disagree. Neither of us is going to convince the other." — T Clark
In a quantum reality we can only talk about it's velocity when measurements were made
— keystone
we were talking in terms of Calculus, and that is a very integral and important circumstance to my question. Perhaps I should have pointed that out. — god must be atheist
You mean the continuum is everything. That is the opposite of nothing. Then what you call continua are the line segments that are fall inbetween these two complementary extremes. — apokrisis
Then what you call continua are the line segments that are fall inbetween these two complementary extremes. — apokrisis
if the line is cut, then you are also talking about a lack of line with some infinitesimal length, not a 0D point. — apokrisis
This just helps show that the idea of a 0D point is ontically problematic and in need of much better motivation than you are providing. You assume too much without providing the workings-out. — apokrisis
Nothing and everything are really the same. A void and a plenum are either too empty to admit change, or too full to admit change. White noise is both every song ever written, or that even could be written, played all a once, and no song being played at all. — apokrisis
continua must exist as a constraint on a state of everything. — apokrisis
Sure. Behind it all is symmetry and symmetry breaking. Numbers are based on the maximum symmetry that is their identity operation - 0 for addition, 1 for multiplication. This first step suffices to produce the integers. Then more complex algebra gives you further levels of symmetry to populate the number line more densely with other symmetry breakings.
There are generators of the patterns. You start with the differences that don’t make a difference. Then this yields a definition of the differences that do.
Again the logic of the dialectic and the basis of semiotics. Stasis and flux are a dichotomy. Mutually dependent and jointly exhaustive. Each is the measure or the other. — apokrisis
To use the usual example, when you say x=0, are you talking about 0.00…. to some countable number of decimal places. Have you excluded x=0.0000….a gazillion places later …0001? — apokrisis
No, I don't believe the continuum is everything. I think that the computer/mind lies outside the continuum. For example, when you imagine a sphere your mind exists outside that sphere. — keystone
In our mind, we are neither thinking of everything nor nothing. We can only think of something. I don't believe in the existence of either extreme. — keystone
When a line is cut, none of the line is lost. It is just divided. — keystone
You're the first to ever entertain my idea on cutting a continuum. (or perhaps you have the same idea) — keystone
The part I have trouble with is your use of 'everything'. I think your 'everything' is every 'potential' thing. My 'everything' is every 'actual' thing (which doesn't include objects/events that don't exist/happen). — keystone
What exactly do you mean by this? I don't think 'a state of everything' needs to exist for 'something' to exist. — keystone
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
How do we recognise the discrete except to the degree it lacks continuity. — apokrisis
The final requirement for a notational system is semantic finite differentiation; that is, for every two characters K and K' such that their compliance-classes are not identical, and every object h that does not comply with both, determination either that h does not comply with K or that h does not comply with K' must be theoretically possible. — Goodman, Languages of Art
So not necessarily a matter of degree. Arguably a matter of discrimination. — bongo fury
I am talking about how the spectrum — apokrisis
that allows your 50 shades of grey — apokrisis
This is confusing for sure. — apokrisis
But after the separation of the potential, you get the new thing of the possibility of a mixing. — apokrisis
So we start with a logical vagueness - an everythingness that is a nothingness. — apokrisis
We have a “greyness” in that sense. Something that is neither the one nor the other. Not bright, not dark. Not anymore blackish than it is whitish. You define what It “is” by the failure of the PNC to apply. You are in a state of radical uncertainty about what to call it, other than a vague and uncertain potential to be a contextless “anything”. It is not even a mid-tone grey as there are no other greys to allow that discriminating claim.
But then you discover a crack in this symmetry. You notice that maybe it fluctuates in some minimal way. It is at times a little brighter or darker, a little whiter or blacker. Now you can start to separate. — apokrisis
You can extrapolate this slight initial difference towards two contrasting extremes. You can drag the two sides apart towards their two limiting poles that would be the purest white - as the least degree of contaminating black - and vice versa. — apokrisis
Once reality is dichotomised in this fashion, then you can go back in and mix. You can create actual shades of grey by Goodman’s approach. — apokrisis
First, there are two different notions of 'the continuum'. One is that the continuum is the set of real numbers R. The other is more specifically that the continuum is R along with the standard ordering on R, or formally the ordered pair <R L> where L is the standard 'less than' ordering on R. — TonesInDeepFreeze
where can one read of a notion of the real continuum as an "n-dimensional continuum"? What does it mean? — TonesInDeepFreeze
where can one read of a notion of the real continuum as an "n-dimensional continuum"? What does it mean? — TonesInDeepFreeze
Suggestion: Since you are interested in formulating an alternative to infinitistic mathematics, then you would do yourself a favor by first reading how infinitistic mathematics is actually formulated, as opposed to how you only think it's formulated, and also you could read about non-infinitistic alternative formulations that have already been given by mathematicians. — TonesInDeepFreeze
If by "Cantor's nonsense" you mean his religious beliefs, then it is plain, flat out false that axiomatic infinitistic mathematics implies Cantor's religious beliefs. — TonesInDeepFreeze
I've heard people say that the paradoxes entwined with actual infinities are beautifully mysterious...I just think they demonstrate the flaws of the concept of actual infinity.
— keystone
What specific paradoxes do you refer to?
Keep in mind that no contradiction has been found in ZFC. — TonesInDeepFreeze
I know, but as usual you don't see where I'm coming from. — bongo fury
...and with those abstract nouns. — bongo fury
Do you mean, an indiscriminate application of colour words to the domain of things (or patches)? — bongo fury
Do you mean, you are able to apply the words in a manner that begins to distinguish two different though still overlapping colours? — bongo fury
Goodman's approach is concrete and clear. Yours is abstract and poetic.
A discrete classification in no way has to imply a continuous one. — bongo fury
Fine. But it's not easy to axiomatize real analysis that way.
One can philosophize all day about how one thinks mathematics should be. But other folks will ask "What are your axioms?" They ask because they expect that an alternative mathematics should have the objectivity of set theory, which is utter objectivity in the sense that, by purely algorithmic means, we can definitively determine whether a purported proof is actually a proof. — TonesInDeepFreeze
Is it possible for a continuum to exist and be defined mathematically without relying on numbers? — keystone
I'm referring to a curve (1D continuum), surface (2D continuum), etc. — keystone
The hotel simply has actually infinite rooms. Do you think it's a gross misrepresentation of infinite sets? — keystone
A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum.
— keystone
As I mentioned, that is not how it is done. You would do yourself a favor by reading a good textbook on the subject so that you would have a basis to critique the actual mathematics rather than what you only imagine is the actual mathematics. — TonesInDeepFreeze
Keep in mind that no contradiction has been found in ZFC.
— TonesInDeepFreeze
Most notably Hilbert's paradox of the Grand Hotel, but also the following:
Gabriel's horn
Galileo's paradox
Ross–Littlewood paradox
Thomson's lamp
Zeno's paradoxes
Cantor's paradox
Dartboard paradox — keystone
The existence of the set of natural numbers is derived axiomatically. Granted, the key axiom is that there exists a successor inductive set, which is an infinitistic assumption. — TonesInDeepFreeze
On the other hand, the notion of "potential infinity" demands alternative axioms.
Take just the non-infinitistic axioms of set theory. What axioms does the "potential infinity" proponent add to get real analysis? — TonesInDeepFreeze
I feel like you could give me a little more slack here on my phrasing. — keystone
you would never try to provide an infinite list of points to completely describe a line (Cantor)
— keystone
Cantor doesn't do that. In fact, Cantor proved that that CAN'T be done. It's his MOST famous result.
You have it completely backwards.
What articles have you read about Cantor that have led you to your terrible misunderstandings? — TonesInDeepFreeze
IF there is any merit to my view, then the hard work hasn't even begun. — keystone
I should have explained explicitly what I meant when I wrote "(Cantor)" as you interpreted my intention backwards. — keystone
If you want to argue for potential infinities over actual infinities, then the real world is surely the better place to test your case.
Arguing against maths using physicalist intuition becomes Quixotic if maths simply doesn’t care about such things. Physics at least cares. — apokrisis
What I have said is that - as the history of metaphysics shows - there are two camps of thought about the physical world. Broadly it divides into the reductionism of atomism and the holism of a relational or systems approach. — apokrisis
You can claim to have no problem with an infinity of cuts and yet have a problem with an infinity of points. — apokrisis
I would say the 0D point and truncated interval are in the same class of question-begging objects. Both are atomised entities lacking a properly motivated existence. — apokrisis
Okay, I accept that substance (continua) and void (0D points) and are both fundamental! — keystone
The problem here is that the real number line is the mathematical object that was in question, surely? So as a construction, it hosts both the rational and the irrational numbers as the points of its line. — apokrisis
And so the claim becomes that reality has a fundamental length – the unit one interval. — apokrisis
Consider the proof that sqrt(2) is an irrational number. I would argue that the proof only demonstrates that sqrt(2) is not a rational number and that something beyond rational numbers must exist on the real line. It does not prove that sqrt(2) IS a number. — keystone
In QM we have come to accept a certain level of uncertainty. Why can't we do the same in math? — keystone
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