Can "Infinity" ever be more than a mathematical abstraction?

Discussion in 'Physics & Math' started by Seattle, Jun 24, 2018.

  1. Speakpigeon Valued Senior Member

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    Let's assume there was already an infinite three-dimensional space in place before the Big Bang, empty or not. When the Big Bang occurs, the initial period of inflation creates all of the space necessary for the bit of universe that we know and inhabit to come out and expand into. No problem whatsoever. Maybe there are other scenarios conceivable, but that's certainly one.
    It's not exactly the principle of the Hilbert Hotel. It's an infinite hotel tower to which you add a few levels at ground level, enough of them, or more, to accommodate the new guests. Guests already in the hotel don't have to move. They don't even notice anything, at least initially and for quite some time.
    EB
     
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  3. Speakpigeon Valued Senior Member

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    Maybe lemurs just do not use or need symbolic representation, I just wouldn't know, but you, you just didn't understand what I just said. You should just take more time to reflect on what people say to just make sure you understand before making remarks showing you just didn't understand what you just commented on.
    EB
     
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  5. Speakpigeon Valued Senior Member

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    I think I'm pretty good at being consistent. I don't think I could be more concise without being obscure, and I try my best to make clear sentences. If it's not good enough for you, that's just too bad.
    EB
     
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  7. someguy1 Registered Senior Member

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    The apple in the air is the thing itself, the territory.

    What is the map? Well, Galileo worked out that the apple acts as if it's drawn to earth with an acceleration of -32ft/sec^2. Newton worked out his universal law of gravitation. Einstein presented general relativity and his new theory of gravity. In each case we have a historically contingent map, each of which is a "pretty good" description of the behavior of the apple, but none of which are the true or ultimate cause of the apple's behavior. Reality just is. Science is a historically contingent human endeavor that provides successively better explanations of what we can observe, up to the limits of our experiments.


    Tallying was probably the world's first example of mathematical abstraction. Og the caveman killed a mastodon, made a mark in the ground. Later on Og killed another mastodon and made another mark in the ground. The association of tallies and dead mastodons is mathematical abstraction. Tallying has been around for what, 100,000 years at least, when we lived in caves.

    Set theory, on the other hand, wasn't invented till the 1870's, not put into its current form till fifty years later in the 19020's, and today goes far far beyond anything that refers to the physical world.

    Now of course set theory in its naive form does tell us about how collections behave. Unions, intersections, one-to-one correspondence, and the like, are useful ideas in the real world. But set theory goes very far beyond that to infinitary reasoning that has no reference to the physical world. So no, to answer your question, I do not believe modern set theory tells us much about tallying. The naive set theory of finite sets perhaps does, but modern theory begins with infinite sets and does not refer to the real world. As far as we know, of course. All human knowledge is historically contingent.

    Yes of course, I agree. But again you are talking about finite combinatorics, not modern set theory.

    Again, finite combinatorics. Not even taught in set theory class. Set theory begins with infinite sets and goes far beyond, to study axiomatic systems whose very meaning even in the world of abstractions is far from clear, let alone the real world. To give you a sense of the modern set theory I'm talking about, take a glance at this Wiki article on large cardinals.

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

    That's what set theorists study. It's not tallying and it's not finite combinatories.

    Finite combinatorics again. I'm perfectly willing to agree that finite combinatorics refers to the real world.

    Set theory? Not a bit of it.

    ps -- I might have been more clear originally if I'd carefully distinguished modern set theory from basic finite combinatorics and tallying. We'd then be in basic agreement that finite combinatorics tells us something real about the world; whereas with modern set theory, the question is at the very least highly debatable.

    pps -- On the other hand I raised my doubts using the examples of the empty sets and singletons, which are part of basic, elementary set theory. So perhaps instead of invoking higher set theory I should stick with the basics.

    Do you think the empty set refers to something in the physical world? It's a good question, there's no right answer. Similarly, if we both agree that there's an apple sitting on teacher's desk, is the set containing the apple sitting on the desk as well? I say no. Only in abstract set theory does such a thing exist. Not in the physical world.
     
    Last edited: Jul 7, 2018
  8. Write4U Valued Senior Member

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    Boy are you a dosagreable fellow
    What is this obsession with the word "just" which I used in proper context. Your attitude is just taxing, to say the least.

    And perhaps you should follow your own advice, and make sure you understand what people say before making remarks you JUST didn't understand what you commented on.
    Moreover you refuse to read any of the links offered for clarification of what is being said with some lame excuse that if you cannot challenge the author it is of no use to you.
    That's JUST bizarre. Moreover, if my posts are not good enough for you, that's JUST too bad.
     
    Last edited: Jul 8, 2018
  9. someguy1 Registered Senior Member

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    727
    I just happened to run across this quote. It's in a long famous discussion thread about the infinite hat problem, which I won't go into here but is a fantastic illustration of the nonsensical but nonetheless logically rigorous and strangely compelling world of abstract mathematics.

    If you have a spare five or ten minutes, read the article. And if you want to see some smart people arguing about some very strange things, read the associated comment thread. Perhaps understanding the infinite hat problem might shake people's seemingly religious faith in the reality of mathematics.

    https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/

    Also see the excellent Wiki article. https://en.wikipedia.org/wiki/Hat_puzzle

    In one of the comments I found a quote from a mathematician named Tom Leinster. He's famous as a mathematician though not known to the public. He doesn't have a biographical Wiki page but there is a Wiki page about something named after him. That pretty much quantifies his level of fame.

    He's a very prominent guy in category theory, which is one of the new alternative foundational approaches that may well supplant set theory in the future.

    In this thread, people are arguing over whether the axiom of choice is "true" or not, and whether the question is even meaningful. Leinster says:

    I’m not an “agnostic” or a “believer” or a “non-believer” in the Axiom of Choice. That’s because it makes no sense to me to ask whether it’s “true”. That would imply some kind of external reality in which sets – including the uncountable ones – are supposed to live.

    If a physicist or astronomer could locate such a reality, then we could have a meaningful debate about whether AC is true. Otherwise, sets are a figment of our imagination – just a formal system. The analogy with the parallel postulate is a good one: it makes no sense to ask whether that’s “true”. There are models where it holds, and models where it fails. What more is there to say?

    To be honest, I’m baffled by talk of a “real world” of sets. I have no idea what it means.


    Now that is exactly how I feel about the matter as well. Mathematical structures are not real. Tegmark is a mathematical Platonist and I am not and neither is Tom Leinster. I recently saw a similar opinion expressed by Joel David Hamkins, a superstar set theorist, speaking in a video. He said that he doesn't know whether it all means anything or not. And if Hamkins doesn't know whether set theory's real, nobody does.

    So all we're talking about I believe is mathematical Platonism. Whether you think the structures of mathematics (including the Hilbert spaces used in QM) are "out there" or not.

    https://plato.stanford.edu/entries/platonism-mathematics/
     
    Last edited: Jul 8, 2018
  10. iceaura Valued Senior Member

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    30,994
    There's also the aspect of continuous vs discrete.
    If the universe is continuous in time and/or space we have physical manifestations of at least some infinities, such as the mentioned series that equals 2 (1+1/2+1/4 - - - ), all around us. Zeno's arrow crosses no gaps, makes no sudden jumps. That is not exactly Platonic.
    If it isn't, then the question of whether "2" exists out there comes forward: if it does, then one would think an infinite series that is equal to it does as well. That would involve Plato.
     
  11. iceaura Valued Senior Member

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    30,994
    But there is a complication. Mathematical structures may be our only way of perceiving certain "things" - something akin to our color vision being our way of perceiving wavelengths within that narrow range.

    Is the color "red" real? It seems to belong to a different category than dragons and unicorns. And other such perceptions - weight, shape, texture, size, etc - we hesitate to discard as not real. But then the mathematical description of an ultraviolet "color" would seem to attain the same status.
     
  12. Write4U Valued Senior Member

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    Hameroff and Penrose believe that the brain does not perceive the color red as a single color at all, but the quantum wave collapse of three wavelengths, each with a specific value, which when processed and combined, produce the illusion of red in our brains.
     
  13. Write4U Valued Senior Member

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    Let's not, no one else does.
     
  14. Seattle Valued Senior Member

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    Maybe you will find this helpful...
     
  15. NotEinstein Valued Senior Member

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    Please give the values in nanometers of the three wavelengths that end up making the color red.
     
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  16. Speakpigeon Valued Senior Member

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    Mathematics is abstract. It's not even meant to refer to anything concrete in the physical world, such as a particular set of apples or a particular set of cardinals. In that, maths works very much like a language does.

    Words, most of them at least, are not thought of a referring to any concrete thing. The word "apple" doesn't refer to any concrete thing, such as a particular apple. However, speakers can obviously use words to refer to concrete objects. Like in the utterance, "Give me this apple" or, "Give me those five apples". Yet, language is getting more like mathematics if we say instead, "Give me one apple" or "Give me five apples". These are very concrete utterances that people make everyday around the world, yet, in these two last examples, "one apple" and "five apples" don't refer to anything concrete like a particular apple or a particular collection of five apples. Instead, they specify what is expected in abstract terms so that the speaker leaves it to the discretion of the hearer to decide which particular apples he will pick to make up one apple or five apples. In the end, the speaker will end up with one apple or five apples, as expected, and yet without ever specifying which particular apples to wanted. This way of doing things obviously works very well. It's very concrete. It's an ordinary, everyday situation not involving highfalutin mathematicians and yet it involves the use of very abstract language.

    This demonstrate I think that abstraction is at the heart of our relation with the physical world and that it's no going to go away.

    So, no, to answer your question, the empty set doesn't refer to anything concrete. Yet, we can use it to communicate just as we routinely use the expression "five apples", which also doesn't refer to anything concrete, to communicate about the physical world, and specify what we want. And it works.

    As I see it, it's just a category error to think of the empty set as meant to refer to something. Words don't refer. Not by themselves. We can use them to refer to something but then we'll use a complete description of what we mean. If I say, "I want those five apples", it specifies clearly, at least in context, what I want. So, it's me in this context who is referring to five particular apples.

    The notion of empty set can be used to specify for example the result of a process as having produced nothing. So, it works much like the word "nothing". And like nothing, there's just one. It's a mistake to talk of "empty sets" in the plural. There's just one, even if in practice we may use the plural. Same thing for "two" or "one thousand". There's just one number 2 and there's just one number 1000. This is because these notions are indeed all abstract and so an empty set of apples is the same thing as an empty set of dinosaurs, just as the number of apples in a pair is the same as the number of dinosaurs in a pair.

    So, if the empty set isn't the kind of thing that could refer, do we have empty sets in the physical world? Well, in the same sense as we have sets of apples and sets of dinosaurs. A set of apple is nothing beyond the apples themselves considered collectively. Since we certainly consider that each apple exists as a concrete, physical object, we're left we only the option of considering that the set of apples doesn't exist as a set. All we have is the set of apples if by that we mean a particular collection of apples.

    I think this shows we can't separate our consideration of abstractions, such as sets and numbers, from how we use them in practice, which is to communicate about the physical world. Yet, the job of mathematicians is precisely to forget about that and consider mathematical abstractions independently of how we might get to use them. It's their job and it seems they are good at it.

    I think the problem only arrise when we try to think of mathematical concepts, such as sets and numbers, but that's true of all mathematical concepts, as possibly referring to something concrete. The number 2 just doesn't refer. We can use it to refer concretely to two particular apples or more abstractedly to any two apples. But 2 doesn't refer. There's no concrete thing that's a 2 in the physical world. Yet, we all understand what people mean when they talk of there being two apples in one basket. Similarly, we can use the empty set to talk about there being no apples in this one basket. We all understand, I hope.

    So, obviously, if we understand and are able to act on what we understand, this suggests that there's something real in the physical world that somehow corresponds to the abstract notions we use to talk about it. The very abstract notion of apple can be used to refer to a very concrete object, there on the table in front of me, and we will all understand what it is. Yet, the notion of apple is abstract. The notion is no particular apple. It's not concrete. It's nothing physical. It's not physical yet we can't stop using it. We effectively need abstract notions to talk about the physical world.
    EB
     
  17. Speakpigeon Valued Senior Member

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    1,123
    (follow up from previous post)

    So, where is the problem? There's no abstract notion of set in the physical world just as there's no abstract notion of apple.Yet, we can talk and understand each other when we talk about a particular set of five apples.

    One way certainly to understand this is to make the distinction between our mental representation of the world and the world itself. The map and the territory. If we stick to this distinction, I fail to see where would be the problem. The notion of sets like the notion of apples are part of our mental representation of the world. When I ask for five apples, I'm expecting the shopkeeper to give me five apples just because I happen to have a representation of the shopkeeper in my mind. I should say that the shopkeeper is itself a mental representation, hopefully of something real existing in the physical world. And I happen to believe the shopkeeper will understand what I mean by five apples, and that will presumably be because the thing in the physical world that hopefully corresponds to my idea of a shopkeeper also has broadly the same mental representation of the physical world as I do.

    So what there is in the physical world doesn't really matter here. What matters is that the communication process should be effective. I want five apples. I don't care whether there is something ontologically real in the physical world that would somehow be five apples. As long as I end up apparently eating those five apples, I'll be happy. So, being pragmatic about this sort of problems certainly helps.

    Things certainly get more complicated when we come to the question of infinities. If I were to ask my shopkeeper to hand me one infinite number of infinitesimally small parts of one apple, he won't possibly know what to do, even if he's a very bright shopkeeper. However, mathematicians and physicists in particular are trying to work out how to use exactly this notion of infinity. We're obviously not quite there yet since they can disagree quite vehemently about it. So, I think we should just take anything said about infinities with a pinch of salt. It's indeed all speculative.

    Yet, as long as we can't decide that infinities definitely don't exist in actual fact, I think it would be a mistake to ask people not to use the concept as if infinities existed. It's impossible to take the necessary linguistic precautions all the time to signal that we know we're talking about something highly speculative that maybe doesn't exist. We have to talk as if we meant business and as if infinities really existed. And as long as we don't meet with any observation contradicting the existence of infinities, it should be taken as legitimate.

    Sorry for this long piece but I was hoping to avoid incomprehension as much as possible given that it's a difficult subject to discuss.
    EB
     
  18. Write4U Valued Senior Member

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    Sorry, red is a primary color and has a single wavelength. My bad.

    I should have been more careful and presented it this way;
    and

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    Close-up of a trichromatic in-line shadow mask CRT display, which creates most visible colors through combinations and different levels of the three primary colors: red, green and blue
    https://en.wikipedia.org/wiki/Trichromacy

    In support of Hameroff, if we consider perception of color intensity being a form of quantum wave collapse, we can begin to understand that very bright colors actually produce pain.
     
    Last edited: Jul 8, 2018
  19. NotEinstein Valued Senior Member

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    Better, but you're still wrong. You've even demonstrated that yourself in your own post by quoting "... three bands of light: long wavelengths, peaking near 564–580 nm (red); ..." Red isn't a single wavelength; it's an band (interval of wavelengths).
     
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  20. TheFrogger Banned Valued Senior Member

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    Actually there is some debate as to what the primary colours are. Photographers disagree...
     
  21. Write4U Valued Senior Member

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    The thrust of my post was not to discuss the properties of wavelengths of colors, but Hameroff's view on our processing of wavelengths of colors in the brain, i.e quantum wave collapse ;
    Trichromacy or trichromatism is the possessing of three independent channels for conveying color information, derived from the three different types of cone cells in the eye.
     
  22. NotEinstein Valued Senior Member

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    I know that, but I spotted a mistake you made, and I corrected you. You then made another mistake, and I corrected that as well. That was all. Carry on.
     
  23. Write4U Valued Senior Member

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    Thanks, I appreciate your interest.
     

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