What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to? — TonesInDeepFreeze
The mathematical definition is given in topology. How could the actual mathematical definition not be at the very heart of comparing the mathematical definition with alternative definitions? It seems to me that you're rationalizing your unwillingness to inform yourself on the subject. — TonesInDeepFreeze
What? You didn't immediately apprehend that was a spoof? — TonesInDeepFreeze
but you haven't the slightest inclination to even glance over a mathematical definition given to you by a person who has, at extreme length and in extreme detail (in at least two other threads) engaged your notions. Why is that? Could it be in your personal characteristics? (Some variation of being so overly infatuated with your own mind that there's little intellectual juice left in you to bother learning much about the mathematics that other people have given lives of intellectual labor to?) — TonesInDeepFreeze
What? You started your post by agreeing that it is an adjective. It is an adjective, a predicate in this case. — TonesInDeepFreeze
But in this discussion, we see people refer to both 'the continuum' and 'continua', so we should be careful not to conflate those terms. — TonesInDeepFreeze
So you have no objection to the axiom of infinity itself, only with philosophizing that there exist "actual" infinite sets? And what do you mean by "actual"? If one views mathematical sets to be mathematically actual but one does not opine as to whether there are physically actual sets, is that okay with you? If one holds that abstractly there are infinite sets but one does not opine that physically there are infinite sets, is that okay with you? — TonesInDeepFreeze
Meanwhile, it seems that the point of my parody went past you — TonesInDeepFreeze
It is to be vigilantly mistrusted. — TonesInDeepFreeze
Who do you think they are comprehensible to, other than yourself? — TonesInDeepFreeze
I'll look at this later, if my time, patience and supply of snacks is adequate. — TonesInDeepFreeze
Why is it that the intro to calculus/analysis textbooks I’ve read never mention topology? — keystone
Thanks for your response. The above video is very interesting but it's minute 2 I'm concerned with. This is how i see all geometric objects, and all objects in general actually. — Gregory
It's not as if i recoil in horror before matter itself, — Gregory
but i don't understand why something in mathematics so simple cannot be explained to me as if I were 8. — Gregory
Maybe I'm just neurally divergent. I've teased apart the finite from the infinite in an object, and in putting them together I find them contradictory, as have many philosophers in history, Hegel being one of them. Good day — Gregory
What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to?
— TonesInDeepFreeze
For example:
That set theory is about infinite sets.
That Cauchy sequences are infinite sequences.
That reals are numbers in the same sense that rationals are numbers.
That the Cartesian (and related) coordinate systems lie at the heart of basic calculus. — keystone
The mathematical definition is given in topology. How could the actual mathematical definition not be at the very heart of comparing the mathematical definition with alternative definitions? It seems to me that you're rationalizing your unwillingness to inform yourself on the subject.
— TonesInDeepFreeze
Why is it that the intro to calculus/analysis textbooks I’ve read never mention topology? Is it because these texts don’t need a general definition of continua since they only work on the continuum, whereas topology is needed for a defining continua? — keystone
Keep in mind, you’ve already given me a reading list that I’m just a few pages into. Adding topology isn’t a problem—I’d even prioritize it if it made sense. But I think it’s fair for me to question whether expanding my reading list is really necessary. — keystone
What? You didn't immediately apprehend that was a spoof?
— TonesInDeepFreeze
Ha! No, I didn’t. But I was being honest with my response. There was so much technical jargon that I had no idea what you were talking about, so I asked ChatGPT. That only made me more confused, so I stopped. — keystone
And I told you that. The same goes for topology—I stopped because I’m not informed on the subject, and I told you that too. — keystone
but you haven't the slightest inclination to even glance over a mathematical definition given to you by a person who has, at extreme length and in extreme detail (in at least two other threads) engaged your notions. Why is that? Could it be in your personal characteristics? (Some variation of being so overly infatuated with your own mind that there's little intellectual juice left in you to bother learning much about the mathematics that other people have given lives of intellectual labor to?)
— TonesInDeepFreeze
I think the main reason we're not fully connecting is that I’m not presenting my points in a way that’s suitable for a mathematician, — keystone
I’m not fully understanding some of your points because they’re not framed in a way that’s accessible to a non-mathematician. — keystone
I’ve been using ChatGPT as a tool to help me grasp the more complex ideas, but you discourage that. — keystone
I only have a limited amount of intellectual juice to dedicate to this. — keystone
I am passionate about my idea but I don't think that's the main factor here. — keystone
But in this discussion, we see people refer to both 'the continuum' and 'continua', so we should be careful not to conflate those terms.
— TonesInDeepFreeze
Yes I get that. I've been consistent with this. — keystone
So you have no objection to the axiom of infinity itself, only with philosophizing that there exist "actual" infinite sets? And what do you mean by "actual"? If one views mathematical sets to be mathematically actual but one does not opine as to whether there are physically actual sets, is that okay with you? If one holds that abstractly there are infinite sets but one does not opine that physically there are infinite sets, is that okay with you?
— TonesInDeepFreeze
My view is that the Axiom of Infinity represents an inductive algorithm for constructing the inductive set, which is said to have a cardinality of aleph-0. — keystone
It is to be vigilantly mistrusted.
— TonesInDeepFreeze
But should it be mistrusted in 2-5 years? — keystone
Who do you think they are comprehensible to, other than yourself?
— TonesInDeepFreeze
A mathematician with plenty of patience and an open mind — keystone
so far it's only been ChatGPT... — keystone
I'll look at this later, if my time, patience and supply of snacks is adequate.
— TonesInDeepFreeze
The next bag of Sweet Chili heat Doritos is on me. :P — keystone
It's not as if i recoil in horror before matter — Gregory
Then halfway through all of it suddenly made sense. — jgill
So they're mapping the infinite plane onto a finite disk by projecting it through a sphere — fishfry
I'm not a big fan of matter. How nice it would be to exist without being subject to the vicissitudes of objects - massive, medium size and subatomic - clashing and banging all around you, wantonly careening at you, and roiling inside you without regard for the effect it all has on you. Matter doesn't care at all about me, so why should I respect it? Well, I do respect some of it - nice people, a lovely beach, a perfect avocado, and some jazz records and math books. But most of the rest of it, phooey! One thing for sure, no one ever involved in a head on automobile accident ever said, "Thank the universe for the laws of physics". — TonesInDeepFreeze
For the first half of the semester I had hardly a clue what was going on, while some of my classmates seemed to understand the material. Then halfway through all of it suddenly made sense. After that introduction, when I got into the regular curriculum for the next semester it seemed almost trivial — jgill
What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to? — TonesInDeepFreeze
First off, why do textbooks for courses in U.S. Civics not mention John Locke, William Blackstone, the Federalist Papers, John Marshall or Plessy v Ferguson? — TonesInDeepFreeze
I didn't say you have to study topology to understand the definition of 'continua'. — TonesInDeepFreeze
Next time I'll make it even more outlandish for you so that it is inescapable. — TonesInDeepFreeze
I don't think you're sincere in wanting to communicate. If you were, you would give people the consideration of clearly articulated concepts. — TonesInDeepFreeze
I asked you already: Who do you think it's suitable for? Especially if not for a mathematician, then who? — TonesInDeepFreeze
Anyway, I speculate that the reason you won't read the substantive material in my posts is psychological. You divert to the false claim that the definition I gave is too specialized....You are so busy espousing that you don't read that to which you respond. — TonesInDeepFreeze
How many examples do you need to appreciate that that bot flat out lies — TonesInDeepFreeze
That's not the axiom of infinity! It is nonsense to say that you don't object to set theory by recourse to agreeing not with the axiom of infinity but with something very very different! How stupid do you take people to be? How stupid do you think people are not to see the sophistry you just pulled? You're insulting. — TonesInDeepFreeze
And not evidence that even the most patient and open-minded Bodhisattva of a mathematician wouldn't tell you, "Get back to me when you've worked out some math". — TonesInDeepFreeze
I am working on putting some of your illustrated explanations into actual mathematics. Might take me some time to assemble into a post, hopefully I will finish and post.
But I've been down this road already with you in another thread. I took a lot of time and effort to turn your gibberish into communicative mathematics. Then, all along the way, you revised your idea, so I revised in response, which is fair. But eventually, your proposal came to an impasse of illogic, yet you wouldn't budge and merely insisted on your notions though they had been shown inconsistent. A dead end with you. But maybe this time it could be different. Hope springs eternal. — TonesInDeepFreeze
If I had to guess where you are headed, I might say that taking a continuum (a line,say) as axiomatic somehow you are cutting it into a fine mesh using the S-B Tree — jgill
But how this has a bearing to elementary calculus is a bit foggy. — jgill
Would it be mathematically possible to project an infinite plane unto a "discrete chunk" (to use QM language)? — Gregory
To me this sounds like a contradiction, — Gregory
but "discrete space" seems like a contradiction to me as well. — Gregory
If it's spatial it has parts. Is discrete defined well in mathematics? Again, they use it in QM. — Gregory
What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to?
— TonesInDeepFreeze
Let me restate the examples I mentioned:
Naïve infinite set theory is thought to be about actually infinite sets when I think it is really about potentially infinite algorithms for constructing the infinite sets. (I want to stay clear of axiomatic set theory since I haven't read the required material.)
Cauchy sequences are thought to be sequences of actually infinite terms when I think they are really about potentially infinite algorithms for constructing the infinite sequences.
When we draw a cartesian plot it is thought that there exist actually infinite points in the plot when I think there really are only finitely many continua, each having infinite potential for partitioning. — keystone
First off, why do textbooks for courses in U.S. Civics not mention John Locke, William Blackstone, the Federalist Papers, John Marshall or Plessy v Ferguson?
— TonesInDeepFreeze
...not an American. — keystone
I don't think you're sincere in wanting to communicate. If you were, you would give people the consideration of clearly articulated concepts.
— TonesInDeepFreeze
Mathematicians hold a high bar for clarity. Might it simply be that I'm not a mathematician? — keystone
I asked you already: Who do you think it's suitable for? Especially if not for a mathematician, then who?
— TonesInDeepFreeze
At this point, a mathematician who can piece together informal ideas. At a later point (once I've read more), a mathematician. — keystone
Anyway, I speculate that the reason you won't read the substantive material in my posts is psychological. You divert to the false claim that the definition I gave is too specialized....You are so busy espousing that you don't read that to which you respond.
— TonesInDeepFreeze
I admit that sometimes when it gets too heavy I glaze over the details. But have I really not adequately responded to many of your points in this thread? — keystone
How many examples do you need to appreciate that that bot flat out lies
— TonesInDeepFreeze
It works well sometimes though. I see it moreso as a handy tool to use with caution. — keystone
that's not the axiom of infinity! It is nonsense to say that you don't object to set theory by recourse to agreeing not with the axiom of infinity but with something very very different! How stupid do you take people to be? How stupid do you think people are not to see the sophistry you just pulled? You're insulting.
— TonesInDeepFreeze
I haven't studied axiomatic set theory but I have taken axiom of infinity to mean that there exists an inductive set. Is that not it? — keystone
What I want to reinterpret this as is 'there exists an algorithm to construct an inductive set'. — keystone
I do pla[n] to respond — keystone
And you skipped recognizing that you strawmanned when you said you'd have to study topology to understand my definition. — TonesInDeepFreeze
f you wish to engage me with this, then know that first I need for you to determine what are all the possible configurations and then to say exactly which are a continuum and which are not, as I mentioned. — TonesInDeepFreeze
I wanted to provide a mathematical definition of 'is a continuum'. I find it in topology. People have been knocking around the term 'a continuum' in a math context. So what is a mathematical definition? I provided one. That's the opposite of distraction.
And topology is a study that informs analysis and put analysis in a broad context. Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction? — TonesInDeepFreeze
I provided a quite streamlined definition. If an equivalent definition could be simpler then you're welcome to state it. Less formally:
A topological space C is a continuum if and only if C is compact, connected and Hausdorff.
So I provided definitions of 'compact', 'connected' and 'Hausdorff'. Those definitions depend on the definition of 'a topology' so I provided that definition, which I couched with only these non-logical notions: 'element' (primitive), 'subset', 'power set', 'union', 'pair' and 'binary intersection', which are quite basic notions of mathematics (except for 'element', they also can be defined back to the sole non-logical primitive 'element'). — TonesInDeepFreeze
I felt hurt and turned off by your tone but for some reason in this thread I'm actually quite appreciative of our interactions. — keystone
Please stay tuned. — keystone
You continually conflate math with physics and I continually note that this is a category error. — fishfry
The only reason you don't want math to fully apply to reality is because you suspect a problem with infinite divisibility, right? — Gregory
I read this same argument in Kant recently. He wants mathematics to come from our intuition of the world yet doesn't believe the second antimony must apply to appearance. The only reason you don't want math to fully apply to reality is because you suspect a problem with infinite divisibility, right? — Gregory
Is not 5 yards minus 3 yards 2 yards? Always, forever? Is not 5 feet minus 3 feet 2 feet? I can get smaller and smaller. There is no reason it should end. — Gregory
You want math to apply to the world when they build bridges but won't go all the way, saying instead there is some invisible indeterminate line across which we can't do math. — Gregory
And you say this without a supporting argument. I don't buy it — Gregory
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