• TheWillowOfDarkness
    1.9k


    Mathematical objects do not have isomorphism either, for each is it own particular concept. 2+2=4 is not the same as another, different concept of 2+2=4. One mathematical rule is not another.

    There is no such thing as a ToE because it violates what an account or theory of something does. Each description we give of something, whether it a state which exists or an eternal concept, is singular are and unique. A ToE if formed on the false premise we can give an account of something be an entirely different thing. The very point of a description, theory or definition is it accounts for one specific thing. None of these things are everything, so a ToE will always fail.

    Completeness, if there is anything approaching it, is only defined in a given a specific account. We can have always have a "complete" account in we may fully describe something as we are aware of it, but this will not be exhaustive of everything because there is always another thing; a different rule, another state, a different concept, not given in this description of a thing we know.
  • alcontali
    802
    Mathematical objects do not have isomorphism either, for each is it own particular concept. 2+2=4 is not the same as another, different concept of 2+2=4. One mathematical rule is not another.TheWillowOfDarkness

    For example, "2+2=4" is not identical to "two plus two is four" but these expressions are still isomorphic under translation. That is why the equality operator needs to be defined explicitly as to clarify when we will still acknowledge these expressions as being equal. The idea in math is that expressions can only be unique up to isomorphism. In the physical world, however, we assume that objects can be really unique.

    By the way, in the formalist view, "2+2=4" is a string, i.e. symbolic language only. it is just a string of symbols. It does not represent anything else than that. Seeking correspondence with the physical universe is not the job of mathematics. It is the prerogative of downstream disciplines, such as science, that will institute empirical formalisms, such as experimental testing, to establish such correspondence.

    In mathematics, the symbol "2" and "4" are exclusively Platonic abstractions, i.e. language expressions, that live in their own abstract, Platonic world. The world of natural numbers are a model for arithmetic theory, in a sense that all theorems provable in arithmetic theory are true in the world of natural numbers. Furthermore, the physical universe is not even isomorphic with the Platonic world of natural numbers. From the point of view of mathematics, these two worlds are unrelated.

    The very point of a description, theory or definition is it accounts for one specific thing. None of these things are everything, so a ToE will always fail.TheWillowOfDarkness

    The ToE is a completely hypothetical theory to which we do not have access, and of which the physical universe is a model. An existing model, i.e. collection of true sentences, always has a theory, if only the model itself. In terms of Kolmogorov complexity, the ToE is the shortest possible summary of the physical universe as model:

    In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of the shortest computer program (in a predetermined programming language) that produces the object as output.

    Asserting that the physical universe has no theory of which the representation is shorter than the full details of the physical universe itself, pretty much amounts to claiming that the universe is completely random. This amounts to asserting that any digital representation of the physical universe is an incompressible string.

    Because of Chaitin's incompleteness theorem, there is no proof possible for this view:

    We know that, in the set of all possible strings, most strings are complex in the sense that they cannot be described in any significantly "compressed" way. However, it turns out that the fact that a specific string is complex cannot be formally proven, if the complexity of the string is above a certain threshold. The precise formalization is as follows [...]

    Hence, the situation is rather as following.

    It is not possible to prove that there exists a ToE, because then you would need to produce a copy of it, which is clearly not available. It is, however, also not possible to prove that there does not exist a ToE, because that assertion would be in violation of Chaitin's incompleteness theorem.
  • TheWillowOfDarkness
    1.9k


    There is no isomorphism within the Platonic realm either, each concept is unique.

    The formalisation of 2+2=4 as just symbols is different to the concept of two plus equals four, which is turn different from another concept using the symbols 2+2=4, which is in turn different to a translation of two equals two equals four form one language to another.

    I'm not speaking about a correspondence to the physical realm, but rather the distinction and identity of different concepts or meanings within the Platonic realm. One concept is never another, is not doing the same thing as another. I'm talking about the necessary distinctions of the platonic realm, which render isomorphism incoherent.

    To assign isomorphism in Platonic realm is to tell a falsehood about the distinctions of the Platonic realm. A ToE is impossible because it cannot cross distinction. Whether in the physical or Platonic realm, any proposed ToE is but one distinction of reality. In being the ToE, as opposed to everything else, it necessarily leaves something out. It always fails to cover of something the distinction which are not it.

    Put simply, it does not matter how complex or not a string might be, for in being itself, it is distinct from everything else. The problem isn't given in the particular length or cycles a representation might have or not, it is that the representation is never thing it describes. Full detail is the only description to give, whether we speak of a physical state or something in the Platonic realm. There can be no "shorter strings" of description, derivation form outside concept or formalisms. Any thing, physical or Platonic, can only be given by itself. Our descriptions only give an account of this thing when it describes it.

    This does no imply randomness. It is not, for example, make 2+2=4 random. Since it is given by the concept itself, it is the nature of that instance of 2+2=4 to have this particular meaning. The same is true of every instance of two plus two equals four. The same is true of every concept of translation between two symbolic languages.

    Whether the definitions of the Platonic realm or instances of measurement of the physical universe, there is a reason are present as such: that what each of them are/do. One was never gong to have a world in which an instance of 2+2=4 meant something else than it does. Same for 2x2m pavers one is using in their backyard. If you've got a 2x2 meter paver, it's was never going to be anything else.
  • alcontali
    802
    There is no isomorphism within the Platonic realm either, each concept is unique.

    The formalisation of 2+2=4 as just symbols is different to the concept of two plus equals four, which is turn different from another concept using the symbols 2+2=4, which is in turn different to a translation of two equals two equals four form one language to another.

    I'm not speaking about a correspondence to the physical realm, but rather the distinction and identity of different concepts or meanings within the Platonic realm. One concept is never another, is not doing the same thing as another. I'm talking about the necessary distinctions of the platonic realm, which render isomorphism incoherent.

    To assign isomorphism in Platonic realm is to tell a falsehood about the distinctions of the Platonic realm.
    TheWillowOfDarkness

    You reject a very fundamental notion of the Platonic realm:

    The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

    It is probably also a rejection of the very concept of abstraction.

    Platonic objects are beliefs expressed in language that arise in an abstract world constructed from basic beliefs. It is a core belief in mathematics that such belief objects can be isomorphic. But then again, there cannot be compulsion in matters of belief. Therefore, you do not need to believe it.

    The mathematical way of thinking ultimately always rests on arbitrary, speculative beliefs with no justification possible, as its epistemic domain is staunchly axiomatic. It invariably seeks to strip away (real-world) meaning. In that sense, it is not meaningful either. It does not seek to be necessarily useful either, and it is often probably not. It only seeks to ensure that derived beliefs are provable from basic beliefs. Hence, at best, it is consistent.

    A ToE is impossible because it cannot cross distinction. Whether in the physical or Platonic realm, any proposed ToE is but one distinction of reality. In being the ToE, as opposed to everything else, it necessarily leaves something out. It always fails to cover of something the distinction which are not it.TheWillowOfDarkness

    If we limit the ToE to a compressed digital version of the physical universe, then Chaitin's incompleteness theorem insists that you cannot exclude that it may exist. Such digital file may not leave out anything that would be considered relevant.

    The problem isn't given in the particular length or cycles a representation might have or not, it is that the representation is never thing it describes.TheWillowOfDarkness

    Even though I agree that a map is not the territory, depending on what you use it for, the map may not need to be the territory.

    Any thing, physical or Platonic, can only be given by itself.TheWillowOfDarkness

    Yes, but according to the formalist philosophy, a Platonic object is its representation. The number 12 is just the string "12". It is equal to itself up to isomorphism. Platonic objects are language expressions only. In that sense, they are different from physical objects, which can consist of matter, energy, and so on.

    Our descriptions only give an account of this thing when it describes it.TheWillowOfDarkness

    Yes, but what is the description of a description if not the description itself?

    (essentially unique up to isomorphism ...)
  • Wayfarer
    8.8k
    Mathematics never says what physics should be talking about.alcontali

    I thought of this exchange when I read about this disovery - a mathematical discovery, by physicists, for which approval was sought from Terence Tao, the world-leading mathematician.
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