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.

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? 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.

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. 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.

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, and I’m not fully understanding some of your points because they’re not framed in a way that’s accessible to a non-mathematician. I’ve been using ChatGPT as a tool to help me grasp the more complex ideas, but you discourage that. To be fair, my difficulty responding to your posts on other threads is what motivated me to start studying logic on my own. However, given life's complexities, I only have a limited amount of intellectual juice to dedicate to this. That said, I’m doing my best. But I understand if you decide that we cannot have a worthwhile discussion.

Note: I am passionate about my idea but I don't think that's the main factor here.

What? You started your post by agreeing that it is an adjective. It is an adjective, a predicate in this case. — TonesInDeepFreeze

I agreed with the meaning you intended but not with the exact words you used. That’s my point. I noticed a flaw but filled in the gap to keep the conversation moving. 'Is a continuum' is indeed a predicate, and 'continuum' is a predicate noun. If you had written 'is continuous,' then 'continuous' would be a predicate adjective. But 'is continuous' is neither a verb nor an adjective.

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.

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. I don’t believe the inductive set itself exists; instead, the inductive algorithm is all we need, and it carries the same cardinality. When someone invokes the Axiom of Infinity, they’re really presenting the inductive algorithm, which is what I believe in. No one has ever exhibited an actual inductive set abstractly in the way that a finite set can be exhibited abstractly. And to be clear, I’m not referring to its physical existence.

Meanwhile, it seems that the point of my parody went past you — TonesInDeepFreeze

I must admit that it did.

It is to be vigilantly mistrusted. — TonesInDeepFreeze

But should it be mistrusted in 2-5 years?

Who do you think they are comprehensible to, other than yourself? — TonesInDeepFreeze

A mathematician with plenty of patience and an open mind...so far it's only been ChatGPT...

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

https://m.youtube.com/watch?v=Jl-iyuSw9KM&t=235s&pp=2AHrAZACAcoFCkNjYyBzYWJpbmU%3D

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. It's not as if i recoil in horror before matter itself, but i don't understand why something in mathematics so simple cannot be explained to me as if I were 8. 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

Why is it that the intro to calculus/analysis textbooks I’ve read never mention topology? — keystone

Over the years colleges have designed their curricula to suit the levels of abstract thought students can bring to the classroom. Calculus is taught in high schools at minimal levels of sophistication and during the first year or two at college with a bit more rigor. But even there the emphasis is on understanding applications of the subject. Most students in these classes are not math majors. Analytic geometry is part of this instruction. The student progresses to a higher level at the junior or senior years. At this point they are usually capable of the sort of abstract thinking that underlies calculus in an advanced calculus course, or an introduction to real analysis course. As for topology, I taught an introductory course at the junior/senior levels.

All of the above may not be true at a sophisticated university with high entrance standards.

And the above are generalizations. Individual students may be more or less proficient than I have indicated. For example, a number of years ago a young freshman at my school registered for my advanced calculus course - and received an A (he was also a talented climber and we became friends). Then I have had seniors who barely passed.

(I began college at Georgia Tech in 1954, and was fortunate to be one of the few incoming students who scored high enough on the entrance exam for me to start anywhere I wished, so I was placed in an experimental class of beginning calculus, immediately taught with epsilon-delta precision, with rigorous proofs. 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)

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

Ok, conformal rescaling. Conformal means "angle preserving." So they're mapping the infinite plane onto a finite disk by projecting it through a sphere.

That's an interesting topic. I'm not sure how it relates to the subject of our conversation but if it's meaningful to you, all to the good.

It's not as if i recoil in horror before matter itself, — Gregory

That's good, since matter is all around us.

but i don't understand why something in mathematics so simple cannot be explained to me as if I were 8. — Gregory

Am I failing to explain a mathematical question?

Perhaps you can phrase your question in a sentence or two, clearly, and I'll do my best to explain.

It might be that I'm not quite understanding your question.

As it is, I don't know what mathematical question you are asking.

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

Well let me know if I can answer any specific questions.

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

You said you have no objection to set theory itself but that you object to misinterpretations of it.

The first in your list there is just an observation that the main areas of interest for prominent set theorists concern infinite sets. That's not a misinterpretation of set theory.

The second is just a theorem of set theory. A theorem of set theory is not a misinterpretation of set theory. So, if you approve set theory itself, then you approve its theorems, including the one you just mentioned.

The third is not something I recall ever reading.

The fourth is just an observation that is seen to be true by opening a textbook. It's not a misinterpretation of set theory.

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

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?

Next, my guess is that, yes, analysis textbooks don't usually define 'continua' for the reason you just mentioned. However, as to topology in general, many analysis books do delve into some topology too. Analysis and topology go hand in hand. It's arguable whether it's better to learn topology first for its generality and elegance that applies to analysis (though that lacks analysis as a motivation and source of examples) or analysis first for its motivation and source of examples (though that lacks the generality and elegance of topology). While, of course, both are a gas to study in and of themselves.

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

Ha! Blatant strawman! I didn't say you have to study topology to understand the definition of 'continua'. It's the opposite since I gave you a definition that is self-contained and needs only understanding of a handful of first chapter set theory concepts. That was the very point, as I mentioned it: to give a definition that needs nothing other than that handful of chapter one set theory concepts.

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

Next time I'll make it even more outlandish for you so that it is inescapable. I'll include Madame Chiang Kai-shek, Krusty The Clown, a wig factory in Duluth, Minnesota and a toothpick on the ground as examples, respectively, of a homeomorphism, isometry, generalized convergence and a product space.

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

And I told you that you don't need to know topology to understand the definition. You don't need to have a prior understanding of one single sentence of topology to understand the definition I gave. And you would see that even just by reading the definition itself. But, of course, prophets such as yourself need not bother with the writings of others.

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 asked you already: Who do you think it's suitable for? Especially if not for a mathematician, then who? Who do you think would read your stuff and understand it at any level that qualifies as grasping and absorbing a reasonable version of what you have in your own mind? I don't think you're sincere in wanting to communicate. If you were, you would give people the consideration of clearly articulated concepts. 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. I'd rather stay on the subject of you. Whatever psychology is going on in you that makes you your own special variation of a crank is more interesting than your malformed, ersatz math musing.

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

You're talking about mathematics. Of course discussion then is going to include mathematics.

And, again, the definition I gave requires only familiarity with a few first-chapter, easy concepts in set theory.

You are so busy espousing that you don't read that to which you respond.

I’ve been using ChatGPT as a tool to help me grasp the more complex ideas, but you discourage that. — keystone

How many examples do you need to appreciate that that bot flat out lies, pretends to explain when it is terribly botching the subject, and is incapable of, or unwilling to, check its misinformation, confusion and illogic. It is really really pathetic that you would so seriously louse up the integrity of your study by reference to a source so inimical to truth, understanding and clarity.

And as if it's not enough that the Internet is inundated with cranks, now we have to contend with bots that are cranks!

I only have a limited amount of intellectual juice to dedicate to this. — keystone

You don't ration wisely.

I am passionate about my idea but I don't think that's the main factor here. — keystone

Contrary to evidence.

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

That deserves a fact check, which I'm too lazy now to research.

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

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.

It is to be vigilantly mistrusted.
— TonesInDeepFreeze

But should it be mistrusted in 2-5 years? — keystone

I don't know. It could be magnitudes worse - fully Orwellian and dystopian (if it continues on the path it's now on) or it could be profoundly better - paradisiacal, or anywhere in a continuum ('continuum', nice word that) between. But I'm pessimistic.

But so what? We're not dealing with what it will be. We're dealing with the toxic, pernicious thing it is now.

Who do you think they are comprehensible to, other than yourself?
— TonesInDeepFreeze

A mathematician with plenty of patience and an open mind — keystone

So a hypothetical being. Not any actual mathematician you've ever met? Not even evidence that there is such a mathematician. 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".

And you think a mathematician would not reply in terms that presuppose, at the least, basic undergraduate mathematics?

so far it's only been ChatGPT... — keystone

I can't handle the pathos of that now.

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

Leave it to you to offer the worst junk food.

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.

It's not as if i recoil in horror before matter — Gregory

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".

Then halfway through all of it suddenly made sense. — jgill

Same for me. But it was in 7th grade in sex ed.

So they're mapping the infinite plane onto a finite disk by projecting it through a sphere — fishfry

Would it be mathematically possible to project an infinite plane unto a "discrete chunk" (to use QM language)? To me this sounds like a contradiction, but "discrete space" seems like a contradiction to me as well. If it's spatial it has parts. Is discrete defined well in mathematics? Again, they use it in QM.

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

An incredible paragraph. You're not just a mathematician

That should be: I'm not just a non-mathematician.

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. But how this has a bearing to elementary calculus is a bit foggy. Perhaps Farey sequences to partition Riemann integrals. Just guessing.

I'm going to put 'k-' in front of words to be clear it's your terminology not to be conflated with the usual mathematical usage.

So far, you haven't defined 'is a k-point' and 'is a k-curve', so, at least for now, I'll take them as primitive, with the axiom: No k-point is a k-curve.

It seems that a k-continuum is a certain kind of finite(?), undirected(?), loopless(?) graph whose k-vertices are either k-points or k-curves. And (I surmise) no k-vertex is connected to itself.

You should state definitively what kinds of configurations are k-continua and are not k-continua. Here's what I have so far:


These are k-continua:

one k-curve

one k-curve connected to one k-point

one k-point connected to a k-curve connected to another k-point


These are not k-continua:

a graph in which occurs a k-point not connected

a graph in which occurs a k-point connected to another k-point

a graph in which occur two or more k-curves but at least one of them is not connected to another k-curve (I surmise)

a graph in which occurs a k-curve connected to more than two k-points


Is there a natural number n>1 such that there is no k-continuum such that there occurs n number of k-curves connected to one another? (In other words do you disallow that a k-continuum may have arbitrarily finitely many connections of curves from one to another?)


Are these disallowed from being k-continua?:

a graph in which occurs a k-curve connected to a k-point connected to another k-curve

a graph in which occurs a k-curve connected to another k-curve connected to a k-point


I'm not sure that exhausts all possible configurations. You should figure it out to define 'is a k-continuum'.

What logic do you use? Classical? Intuitionistic? Other? If you don't state your logic then I will take it to be classical.

What set theory axioms do allow? Perhaps Z set theory without the axiom of infinity and the axiom of regularity (though, I predict you will soon enough be stumped without the axiom of infinity). Note: Just saying "finite processes, no infinity" is hand-waving unless you mathematically define 'is a process' without recourse to infinite sets. If you mean Turing machines or equivalents, then let's see how you actually couch your definitions and inferences with them.

For reference, here are the axioms of Z set theory without the axiom of infinity and the axiom of regularity (a symbol guide for the text symbols I use is in my forum About panel):

(1) extensionality:

Axy(Az(zex <-> zey) -> x = y)

For any sets x and y, if they have the same members, then x = y.


(2) separation schema:

If F is a formula in which x does not occur free, then all closures of
ASExAy(yex <-> (yeS & F))
are axioms

If F is formula, then for any set S, there is the subset {x | xeS & F}. Even more informally, for any formalizable property R and set S, there is the set of elements of S that have the property R.


(3) union:

ASExAy(yex <-> Ez(zeS & yez))

For any set S, there is the set of all and only the members of members of S.

Notated as US = {y | Ez(zeS & yez)}.


(4) pairing:

ApqExAy(yex <-> (y = p v y = q))

For any sets p and q, there is the set such that both p and q are in the set and nothing else is in the set.

Notated as {p q}.


(5) power set:

ASExAy(yex <-> y is a subset of S)

For any set S, there is the set whose members are and only the subsets of S. Notated as PS = {y | y is a subset of S}


Summarizing:

If x and y have the same members, then x is y.

For any formalized property and for any set, there is the subset whose members have that property.

For any set S, there is the union of S.

For any sets x and y, there is the set {x y}.

For any set S, there is the set of subsets of S.

(If we add the axiom of infinity and the axiom of countable choice, then we can derive classical analysis. cf. 'Notes On Set Theory' - Moschovakis pg. 116)

For reference, here are definitions of 'is a graph', 'is an undirected graph' and 'is a loopless graph':

G is a graph
<->
EVDf(G = <V D f> &
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(f(b) = {p q} v f(b) = <p q>)))

In other words: a graph is a non-empty set of vertices, a set of edges disjoint from the set of vertices, and a function that assigns to each edge either an unordered or ordered pair of vertices. (An unordered pair of vertices has no direction; an ordered pair of vertices has a direction.)

G is an undirected graph
<->
EVDf(G = <V D f> &
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(f(b) = {p q})))

In other words: an undirected graph is a graph such that there are edges with direction.

G is a loopless graph
<->
EVDf(G = <V D f> &
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(p not= q & (f(b) = {p q} v f(b) = <p q>))))

In other words: a loopless graph is a graph such that there is no edge between any vertex and itself.

Once you satisfactorily cover https://thephilosophyforum.com/discussion/comment/935856 I'll see about rigorously defining 'is a k-continuum' as a graph. But I see trouble ahead anyway with your personal notion of partitioning. I pessimistically predict that ultimately you won't be able to get past handwaving with it.

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

That seems to be the way with a lot of things. I'm certainly hoping to finally reach such moments of clarity in logic, and eventually topology.

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.

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.

I didn't say you have to study topology to understand the definition of 'continua'. — TonesInDeepFreeze

All right, I gave up too soon. I'm not going to be able to respond to everything tonight as I'm running out of time and there's a lot to respond to (which I'm greatly appreciative of). Tomorrow is looking like it will be a busy day for me as well but I do plan to respond to everything. Included in that, I'll spend time trying to understand your topological definition of continua and provide a response.

Next time I'll make it even more outlandish for you so that it is inescapable. — TonesInDeepFreeze

Sadly, sometimes I need people to explain jokes to me....

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?

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.

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?

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.

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? What I want to reinterpret this as is 'there exists an algorithm to construct an inductive set'.

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

Well, that's basically where things ended with you, fishfry, and jgill in the last thread I was active on. That seems to be where things are heading in this thread. I'm starting to get the point.

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

I see you've already followed up on this. I haven't read it yet as I really want to spend sufficient time digesting it and responding. As I mentioned in my last post, my time is short tomorrow so it may be a couple of days before I respond but I do play to respond to that and the topology comments you made in earlier postings. At this point I'll just say a big thanks to you!

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

I think the S-B tree is just one particularly pleasant way to cut a continuum.

But how this has a bearing to elementary calculus is a bit foggy. — jgill

...we never did get to calculus in the last thread. I don't know whether we'll get there in this thread either. Let's see.

Would it be mathematically possible to project an infinite plane unto a "discrete chunk" (to use QM language)? — Gregory

Sure. I'll show the procedure in a moment. But what does that have to do with QM? You continually conflate math with physics and I continually note that this is a category error. Do you mean discrete energy levels? That's not any more mysterious than the discrete natural numbers 1, 2, 3, 4, ...

To me this sounds like a contradiction, — Gregory

I will show the procedure in a moment.

but "discrete space" seems like a contradiction to me as well. — Gregory

Here's the Wiki article on discrete space.

A topological space is discrete if all its points are isolated.

A point is isolated if you can draw a little circle around it that doesn't contain any of the space's other points.

Example of an infinite, discrete space: The integers. Think of the integers on the number line:

......-2...-1...0...1...2...3...4.....................

You can see that around each integer, I could draw a little circle of radius 1/4, say, and that circle would not contain any other integer. So each point of the integers is isolated. And since all of the points of the integers are isolated, we say the integers are a discrete space.

We can make any set into a discrete topological space by simply declaring that every subset of the space is an open set.

Alternately, we can declare the set to have the discrete metric, in which the distance between a point and itself is 0, and between any two distinct points is 1.

With this definition we can make the real numbers into a discrete topological space. I know this is counterintuitive, but it only involves an abstract definition. It's a logic game more than anything else, but it's a fact that we can turn the real numbers into a discrete space simply by giving it the discrete metric.

And then the identity function, which maps each real number to itself, is a function that maps the continuous reals into the discrete reals.

That is, we take two copies of the real numbers. One is given the usual topology, which makes it a continuum; and the other is given the discrete topology, which makes it a discrete space.

Then the identity function from the continuous copy of the reals to the discrete copy maps a continuous set to a discrete set. It's even a bijection, which is extra counterintuitive. But that's one way to do it.

Note that there is no physics involved. This is a purely mathematical exercise.

If it's spatial it has parts. Is discrete defined well in mathematics? Again, they use it in QM. — Gregory

Yes, in math a discrete space is any space where every point is isolated, as I noted.

You might be a little put off by using "space" to mean any old set with some kind of topological structure. It has nothing to do with space as in physics or cosmology. And nothing to do with QM. Just math.

ps -- You asked about mapping an infinite plane onto a discrete space. In that case just do the same trick with two copies of the Cartesian plane, one with the usual Euclidean metric and the other with the discrete metric. It's just a math trick, maybe less to it than meets the eye. But we can definitely map continuous spaces into discrete ones.

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

What is your trip, man?

You said, "my only qualm with set theory is the philosophy underlying it (centered around actual infinities)".

So that would be taken to mean that you have no objection to set theory itself but only to certain philosophy about it. But set theory itself is the set of theorems derived from the axioms. So you would have no objection to the axioms. But you do object to the axioms (and not just philosophy about them). For example, you do object to the axiom that there exists a successor inductive set while instead you approve the very different principle "there are potentially infinite algorithms for constructing infinite sets".

News flash: "there are potentially infinite algorithms for constructing infinite sets" is NOT the axiom of infinity. Saying you approve "there are potentially infinite algorithms for constructing infinite sets" but that you don't approve "there exists a successor inductive set" is NOT saying you don't object to the axiom of infinity but only object to philosophy regarding it; rather it is saying you DO object to the the axiom of infinity but approve a very different principle. Again, you take me for stupid if you think I don't see that. It is a childish game you play.

Same for Cauchy sequences and the Cartesian plane.

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

You're as bad at history as you are at honest, logical thinking. The Federalist Papers, John Marshall and Plessy v Ferguson are American. John Locke and William Blackstone are English but at the deepest foundation of American principles of government and law.

It seems you missed the analogy. You asked why topology is not very much featured in calculus and analysis books you've read. Putting aside that topology is often part of analysis books, the answer is that a calculus book is about setting forth the most basic mathematics that is put to use in different fields of study; it's not about very much understanding the foundations of that mathematics or its broader mathematical context. An analysis book is more foundational than a calculus book but still mainly about diving right into the real and complex numbers as opposed to getting deep into the details of the set theoretic foundations and the topological generalizations.

By analogy, a book for a course in U.S. Civics is mainly concerned with describing the federal government, its organizations and functions, relationship with the states and with its residents and citizens, with some of the philosophical and legal background but not deep into those.

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 think it's more likely that you're a self-infatuated poseur.

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

In other words, a mathematician of a certain temperament and then later a mathematician. Huh?

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

Your idea of a "detail" is a one-liner axiom. You would glaze over just about anything. With all the glazing over you do, you should be in the business of hams or donuts.

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

A quart of raw milk with listeria works well to quench your thirst. A handy beverage to drink with caution.

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

That is it.

What I want to reinterpret this as is 'there exists an algorithm to construct an inductive set'. — keystone

Make up whatever principles you want. But "there exists an algorithm to construct an inductive set" is not the axiom of infinity and it's not a "reinterpretation" of the axiom of infinity. It's a decidedly different idea.

Again, when you say you don't object to set theory but you do object to one of its axioms while preferring to plunk down for a different principle, you make no sense! Would you please stop it? Have at espousing whatever alternative principles you like, but it makes no sense to say you don't object to set theory when explicitly you do!

/

And you skipped recognizing that you strawmanned when you said you'd have to study topology to understand my definition.

I do pla[n] to respond — keystone

If 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.

And you skipped recognizing that you strawmanned when you said you'd have to study topology to understand my definition. — TonesInDeepFreeze

I thought I indirectly addressed this when I said I was going to go back to your earlier messages on topology and respond to them. But if you're looking for something explicit, yes I unintentionally strawmanned you.

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 do wish to engage you with this. On top of being busy these days, I expect it to take me a few days to read, digest, and respond to everything you've said. Please stay tuned.

As an aside, in our previous discussions I felt hurt and turned off by your tone but for some reason in this thread I'm actually quite appreciative of our interactions. Thanks!

Your strawman arose because you don't take seriously (though you make it a point to say that you do) the posts to which you reply.

You suggested that topology might be a distraction and that my definition might be unnecessarily complicated

I replied:

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

and

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

You replied by saying again that topology is not necessary, as if I had said it is necessary, as you were replying to a post in which I said that there's nothing in my definition that requires knowing anything more than a few basic set theory concepts. So, clearly, you ignored what I said and made it appear that I had been saying that you'd have to study topology to track my definition - the opposite of what I said. That is bad faith, whether intentional or a product of not bothering to read.

After a lot of that kind of thing with you, and much worse instances, in this thread and previous ones, I eventually get to the point of feeling that what is most interesting about you is not the particulars of your confusions about mathematics but what goes on in the head of person who is so wrapped up in himself that he wants other people to take their time and labor to understand his own personal, malformed musings while he pretty much just blows off the substantive and informative information and explanations he gets in reply.

The most salient instance lately is your ridiculous claim that you don't object to set theory but only to philosophy about it, yet you do object to the axiom of infinity, while evading that to do so is indeed to object to set theory, not just philosophy about set theory, and then you so sophistically evade that point too by endorsing not the axiom of infinity but a principle very different from the axiom of infinity! Sheesh!

And you claim not to object to set theory but to "misinterpretations" of it, yet your examples of supposed misinterpretations are not misinterpretations! Sheesh and double sheesh!

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

You're a saint.

Please stay tuned. — keystone

The show I'm interested in seeing is a hoped for episode in which you account for all the configurations to determine which are a continuum and which are not. If that goes well, then the studio might consider extending the series into another season.

You continually conflate math with physics and I continually note that this is a category error. — fishfry

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? 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. 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. And you say this without a supporting argument. I don't buy it

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've never known a fellow mathematician who would have agreed with this. A mathematical philosopher perhaps. Let's see what @fishfry has to say. I always enjoy his commentaries.

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

Not at all. Infinite divisibility is not in question within math. But there's no evidence for it in physics. That's my only point.

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

In physics there is a minimum distance, below which we can not sensibly apply our laws of physics. There is no infinite divisibility in physics.

https://en.wikipedia.org/wiki/Planck_units

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

No, there's a measurement boundary below which we can't do physics. And it's not something I say. It's something Max Planck said, and that a hundred years of physics has found no exception to.

And you say this without a supporting argument. I don't buy it — Gregory

You deny the Plank length? I don't follow your point at all. Physics is very clear on this matter. Nature has a minimum length, below which we can't reason sensibly about. That doesn't mean that infinite divisibility isn't part of nature; it only means that infinite divisibility is not a part of our best theories of physics.

That's why I say that infinite divisibility is part of math; but as far as we know, and until some future genius not yet born comes up with a new idea, it's not part of physics.

For me the world is as mathematical as geometric imagery. The world is mystical, nah, miraculous in how it is woven together. Maybe mathematics gives you that sense too. Thanks for the conversation!

The world is mystical, nah, miraculous in how it is woven together. — Gregory

On this we fully agree.

Maybe mathematics gives you that sense too. Thanks for the conversation! — Gregory

Likewise, thanks.