4 ms·

Thanks for writing this up. That actually clears up a lot. So first of all, the kinds of things you are talking about here are way beyond the scope of what I'

by lisper 2y ago

Thanks for writing this up. That actually clears up a lot. So first of all, the kinds of things you are talking about here are way beyond the scope of what I'm writing. This is a series about the scientific method, not about math per se, and the target audience is bright but non-technical high school-age kids. Math is relevant only insofar as it turns out to be extraordinarily useful for building scientific models. Second, the topic of this post was not existence, it was truth. Existence is (mostly) a red herring. Yes, the thing you describe exists. It exists in the ontological category of mathematical objects, which is a sub-category of the ontological category of ideas. That's all I have to say about it. So yes, I could summarize Platonism vs Formalism vs Constructivism, but that is simply not what I'm writing about. The reason I'm bringing up truth in math at all is to lay the foundation for a discussion of whether or not the quantum wave function describes something real. But that won't come until much, much later. BTW, in case you're wondering, the answer is yes, the quantum wave function does describe something real. See: https://blog.rongarret.info/2018/05/a-quantum-mechanics-puzzle.html https://blog.rongarret.info/2018/05/a-quantum-mechanics-puzz... https://blog.rongarret.info/2018/05/a-quantum-mechanics-puzzle-part-deux.html https://blog.rongarret.info/2018/05/a-quantum-mechanics-puzz...

lanstin 2y ago
Do you have an example of a mathematical theory of quantum field theory where the function is not commplete in the sense that makes things uncountable?
- lisper 2y ago
  I'm not sure I understand the question, but I'm pretty sure that the answer is "no". I'm pretty sure that uncountability doesn't actually matter for any practical or even metaphysical purpose, but that's just my personal guess. I have no actual data to support it, and it's mostly above my pay grade. Does that answer your question?
  - lanstin 2y ago
    The difference between a Hilbert space and an inner product space is that a Hilbert space is complete, i.e. has an uncountable number of points. Its all fine and dandy to say that in the history of the universe we can only come up with a finite number of numbers or even finite number of mathematical concepts, but all the current theories of QFT are on top of Hilbert spaces.
    - lisper 2y ago
      OK, but that is a consequence of the continuity of space, right? So you don't even need to go to QM. Basic Newtonian mechanics is all you need to get an uncountable continuum. But space might turn out to be quantized. In any case, the answer to your original question is obviously "no". Why do you think this matters?
      - lanstin 2y ago
        Weirdly it's not related to regular space-time that we seem to inhabit - the Hilbert space in which QM happens does not correspond to the space of special relativity. Space and time of specific things are both the result of applying operators to the Hilbert space. The Hilbert space completeness means you have nice properties as far as orthonormal bases and that sort of thing (venturing outside my knowledge of QM here). The dimensionality of the Hilbert space is related to how many distinct quantum states can exist; it is unknown to physics if the universe has a finite dimensional Hilbert space or infinite. Since starting this conversation, I have tried to find any example of a quantum theory based on something other than the mathematics of the continuum without luck (but I'm not well positioned to find such a thing, as I would guess it would come from some constructivist mathematics thing but even regular ZFC QFT isn't formalized yet).
        lisper 2y ago
        I'm at the hairy edges of my understanding here as well, but AFAIK infinite-dimensional Hilbert spaces arise when modeling position-momentum states because it is assumed that space is continuous.
btilly 2y ago
I see your goal. But I question how you're going about it. First, I'm all for convincing people that the wave function is something real. If you believe that strongly enough, then you're pushed towards accepting the Everett Interpretation. Which makes far more sense to me than any other interpretation of QM. But I would suggest that you get them there by the shortest reasoning that you can. With minimal detours. The more material you have, the more likely that someone with different preconceptions will find something to disagree about. Worse yet, people will lose track and abandon it. I would also suggest that you simplify your writing. According to https://readabilityformulas.com/readability-scoring-system.php https://readabilityformulas.com/readability-scoring-system.p... your writing is college entry level. Readability is weird, we don't notice that it matters until we find ourselves unable to understand text. And then it is a wall. Meaningless phrasing choices can improve readability and broaden your audience. Now for a more substantive criticism. But one of choice of topic, where you likely reasonably disagree with me. You are writing about the scientific method. But the style of your writing undermines the essential spirit behind the scientific method. What do I mean by that essential spirit? I mean the thing that Feynman was trying to explain in http://www.feynman.com/science/what-is-science/ http://www.feynman.com/science/what-is-science/. It is the attitude that leads to statements like, "Science is the belief in the ignorance of experts." Now you're telling them what to believe. You aren't telling them how they could think of it. You aren't encouraging them to disagree with you and question you. You're telling them what is true. I think that conveying the skeptical attitude is more important than convincing them of any particular fact. Moving on, I see you've added a number of things to your article. They are significant improvements. But I would suggest an improvement to one in particular. Your arithmetic example. Counting to negative numbers for debt makes sense. But you don't show how you get something that unexpectedly is useful. So I would make the chain that you lead people down to be counting, to positive integer arithmetic, to negative numbers for debt, to fractions, to the idea that we can put this on a line, to the question of whether we have everything on that line, to the reals. Just outline it. And then point out how we started with counting, and wound up most of the way to geometry...doesn't look like counting any more! Finally, I think that high school students are capable of more philosophy than you do. In particular there are lots of youtube videos and articles that try to explain why uncountable means more. If they can see those explanations, why can't they encounter the idea that all such claims require additional assumptions? Thinking about how to connect reasoning and computation is something of interest. And leads to topics that are fundamental in such places as logic and CS. Anyways, this is wrong and I don't know if you read it. But if you do, hopefully you found at least one thing that's worth thinking about.

by lisper 2y ago

lanstin 2y ago

Do you have an example of a mathematical theory of quantum field theory where the function is not commplete in the sense that makes things uncountable?

lisper 2y ago
I'm not sure I understand the question, but I'm pretty sure that the answer is "no". I'm pretty sure that uncountability doesn't actually matter for any practical or even metaphysical purpose, but that's just my personal guess. I have no actual data to support it, and it's mostly above my pay grade. Does that answer your question?
- lanstin 2y ago
  The difference between a Hilbert space and an inner product space is that a Hilbert space is complete, i.e. has an uncountable number of points. Its all fine and dandy to say that in the history of the universe we can only come up with a finite number of numbers or even finite number of mathematical concepts, but all the current theories of QFT are on top of Hilbert spaces.
  - lisper 2y ago
    OK, but that is a consequence of the continuity of space, right? So you don't even need to go to QM. Basic Newtonian mechanics is all you need to get an uncountable continuum. But space might turn out to be quantized. In any case, the answer to your original question is obviously "no". Why do you think this matters?
    - lanstin 2y ago
      Weirdly it's not related to regular space-time that we seem to inhabit - the Hilbert space in which QM happens does not correspond to the space of special relativity. Space and time of specific things are both the result of applying operators to the Hilbert space. The Hilbert space completeness means you have nice properties as far as orthonormal bases and that sort of thing (venturing outside my knowledge of QM here). The dimensionality of the Hilbert space is related to how many distinct quantum states can exist; it is unknown to physics if the universe has a finite dimensional Hilbert space or infinite. Since starting this conversation, I have tried to find any example of a quantum theory based on something other than the mathematics of the continuum without luck (but I'm not well positioned to find such a thing, as I would guess it would come from some constructivist mathematics thing but even regular ZFC QFT isn't formalized yet).
      - lisper 2y ago
        I'm at the hairy edges of my understanding here as well, but AFAIK infinite-dimensional Hilbert spaces arise when modeling position-momentum states because it is assumed that space is continuous.

btilly 2y ago

I see your goal. But I question how you're going about it. First, I'm all for convincing people that the wave function is something real. If you believe that strongly enough, then you're pushed towards accepting the Everett Interpretation. Which makes far more sense to me than any other interpretation of QM. But I would suggest that you get them there by the shortest reasoning that you can. With minimal detours. The more material you have, the more likely that someone with different preconceptions will find something to disagree about. Worse yet, people will lose track and abandon it. I would also suggest that you simplify your writing. According to https://readabilityformulas.com/readability-scoring-system.php https://readabilityformulas.com/readability-scoring-system.p... your writing is college entry level. Readability is weird, we don't notice that it matters until we find ourselves unable to understand text. And then it is a wall. Meaningless phrasing choices can improve readability and broaden your audience. Now for a more substantive criticism. But one of choice of topic, where you likely reasonably disagree with me. You are writing about the scientific method. But the style of your writing undermines the essential spirit behind the scientific method. What do I mean by that essential spirit? I mean the thing that Feynman was trying to explain in http://www.feynman.com/science/what-is-science/ http://www.feynman.com/science/what-is-science/. It is the attitude that leads to statements like, "Science is the belief in the ignorance of experts." Now you're telling them what to believe. You aren't telling them how they could think of it. You aren't encouraging them to disagree with you and question you. You're telling them what is true. I think that conveying the skeptical attitude is more important than convincing them of any particular fact. Moving on, I see you've added a number of things to your article. They are significant improvements. But I would suggest an improvement to one in particular. Your arithmetic example. Counting to negative numbers for debt makes sense. But you don't show how you get something that unexpectedly is useful. So I would make the chain that you lead people down to be counting, to positive integer arithmetic, to negative numbers for debt, to fractions, to the idea that we can put this on a line, to the question of whether we have everything on that line, to the reals. Just outline it. And then point out how we started with counting, and wound up most of the way to geometry...doesn't look like counting any more! Finally, I think that high school students are capable of more philosophy than you do. In particular there are lots of youtube videos and articles that try to explain why uncountable means more. If they can see those explanations, why can't they encounter the idea that all such claims require additional assumptions? Thinking about how to connect reasoning and computation is something of interest. And leads to topics that are fundamental in such places as logic and CS. Anyways, this is wrong and I don't know if you read it. But if you do, hopefully you found at least one thing that's worth thinking about.