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[…] of compression, universal across effective compressers, in the limit of data." Let me make this more concrete. Imagine you areSo, what is an "effective compresser". By which I mean? Roughly, you are somethingimagine there is an algorithm that receives datapoints (in the "real world", think sense-experience), and you can dothen performs some kind of "operation" to encode them. You also haveThere is time to do sotry different schemes. ExampleLoose example: a scientist who is reasonably intelligent. The scientist observes the world. He seeks to find a "theory" to understand these datapoints, and he has his lifetime to do so. His basic goal is to find methods of"programs" for compression that allow him to parsimoniously "recreate" (these datapoints, which in turn, helps him to predict, them (assuming his methods of compression identify true "hidden causes", and ignoring the problem of induction) these datapoints. Eventually the scientist may, inevitably,will arrive on certain methods ofprograms for compression. Maybe he'll needfind numbers. In fact, and with enough time, he is sure to discover them"something like" numbers. So perhaps, there are indeed methods of compression that are universal or "inevitable" for effective compressers with enough datapoints. This is what I imagine math aspires to. For example, maybe addition is one of those universal methods of compression. What I'm saying, then, here is that if you're trying to compress datapoints, and you have […] like" addition (up to isomorphism?) will be strictly useful to encode. Maybe. And, so any compresser that's good enough and has datapoints, will do so. […] random (I can prove this fact quite easily, if you'd like -- the basic idea is that if you define "not random" as "described with a short program", you run into the problem that there are many more long strings than short strings, so short inputs can cover only a small part of the terrain). But then there is always a computer to which those "random" strings *aren't* random. So theThe extent to which patterns or randomness isare necessarily observer dependent is a genuine facet of math being limited. But furtherEven in the formal world, compression is limited. Further yet, there is a deeper point, which perhaps is related to what you are getting at, which. The point is thatthis: it is not just the question of whether these "datapoints" (left undefined) are patterned (compressible) or not, BUT the fact that […] being affected by them, making sense of them, etc. that is then itself a totally different thing. You can never "complete" this task. You don't get anything whatsoever about firsthand experience from "universals"; in fact, firsthand […] this experience? Maybe not. Some things are defined by their limitations. Not sure if Can anything be defined from the outside? For that matter, can anything be defined from the inside? **Fixing things in formal terms delimits those things, but delimit the world it does not.** I fully appreciate your postam still in synthesis mode, but these are the sparks that were struck for me. Hopefully some are enjoyable.

  1. Current versionHenry D’Angelo·
    Changes from Version 16

    […] complexity, which quantifies the randomness of strings relative to a computer. I'll use finite binary strings. Informally, the Kolmogorov complexity K(x) for a string x is equal […]

  2. Version 16Henry D’Angelo· Compare with the current version
    Changes from Version 15

    […] terms delimits those things, but delimit the world it does not.** Not sure if I fully appreciate your post, I am still in synthesis mode, but these are the sparks that were struck for me. Hopefully some are enjoyable.

  3. Version 15Henry D’Angelo· Compare with the current version
    Changes from Version 14

    […] random (I can prove this fact quite easily, if you'd like -- the basic idea is that if you define "not random" as "described with a short program", you run into the problem that there are many more long strings than short strings, so short inputs can cover only a small part of the terrain). But then there is always a computer to which those "random" strings […] genuine facet of math being limited. Even in the formal world, mathcompression is limited. Further yet, there is a deeper point, which perhaps […]

  4. Version 14Henry D’Angelo· Compare with the current version
    Changes from Version 13

    […] of compression, universal across effective compressers, in the limit of data." Imagine you areSo, what is an "effective compresser". By which I mean? Roughly, you areimagine there is an algorithm that receives datapoints (in the "real world", think sense-experience), and then performs some kind of "operation" to encode them. You haveThere is time to do sotry different schemes. ExampleLoose example: a scientist who is reasonably intelligent. The scientist observes the world. He seeks to find a "theory" to understand these datapoints, and he has his lifetime to […] "programs" for compression that allow him to parsimoniously "recreate" these datapoints, andwhich in turn, helps him to predict them (assuming his methods of compression identify true "hidden causes", and ignoring the problem of induction). Eventually the scientist may, inevitably,will arrive on certain programs for compression. Maybe he'll needfind numbers;. In fact, with enough time, he is sure to discover "something like" numbers. So perhaps there are […]

  5. Version 13Henry D’Angelo· Compare with the current version
    Changes from Version 12

    […] of compression, universal across effective compressers, in the limit of data." Let me make this more concrete. Imagine you are an "effective compresser". By which I mean, you are an algorithm that receives datapoints (in the "real world, sense-experience), and then performs some kind of "operation" to encode them. You […]

  6. Version 12Henry D’Angelo· Compare with the current version
    Changes from Version 11

    […] of them, etc. that is itself a totally different thing. You can never "complete" this task. You don't get anything whatsoever about firsthand experience from "universals"; in fact, firsthand […]

  7. Version 11Henry D’Angelo· Compare with the current version
    Changes from Version 10

    […] limited. Further yet, there is a deeper point, which perhaps is related to what you are getting at. The point is this: it is not […]

  8. Version 10Henry D’Angelo· Compare with the current version
    Changes from Version 9

    […] you are an "effective compresser". By which I mean, you are somethingan algorithm that receives datapoints (sense-experience), and you can dothen performs some kind of "operation" to encode them. You also have time to do so. Example: a scientist who is reasonably intelligent. […] his lifetime to do so. His basic goal is to find methods of"programs" for compression that allow him to parsimoniously "recreate" these datapoints, and in […] problem of induction). Eventually the scientist may, inevitably, arrive on certain methods ofprograms for compression. Maybe he'll need numbers, and; with enough time he is sure to discover "something like" numbers. So perhaps, there are indeed methods of compression that are universal or "inevitable" […]

  9. Version 9Henry D’Angelo· Compare with the current version
    Changes from Version 8

    […] necessarily observer dependent is a genuine facet of math being limited. But furtherEven in the formal world, math is limited. Further yet, there is a deeper point, which perhaps is what you are getting at, which. The point is thatthis: it is not just the question of whether these "datapoints" (left undefined) are patterned (compressible) or not, BUT the fact that […] being affected by them, making sense of them, etc. that is then itself a totally different thing. You don't get anything whatsoever about firsthand […] outside? For that matter, can anything be defined from the inside? Even in the formal world, there are limitations. Ultimately... **Fixing things in formal terms delimits those things, but delimit the world […]

  10. Version 8Henry D’Angelo· Compare with the current version
    Changes from Version 7

    […] outside? For that matter, can anything be defined from the inside? Even in the formal world, there are limitations. Ultimately... **Fixing things in formal terms delimits those things, but delimit the world it does not.** Not sure if I fully appreciate your post, I am still in synthesis mode, but these are the sparks that were struck for me. Hopefully some are enjoyable.

  11. Version 7Henry D’Angelo· Compare with the current version
    Changes from Version 6

    […] outside? For that matter, can anything be defined from the inside? **Fixing things in formal terms delimits those things, but delimit the world it does not.** Not sure if I fully appreciate your post, but these are the sparks that were struck for me. Hopefully some are enjoyable.

  12. Version 6Henry D’Angelo· Compare with the current version
    Changes from Version 5

    […] outside? For that matter, can anything be defined from the inside? Fixing things in formal terms delimits those things, but delimit the world it does not. Not sure if I fully appreciate your post, but these are the sparks that were struck for me. Hopefully some are enjoyable.

  13. Version 5Henry D’Angelo· Compare with the current version
    Changes from Version 4

    […] defined by their limitations. Can anything be defined from the outside? CanFor that matter, can anything be defined from the inside? Not sure if I fully […]

  14. Version 4Henry D’Angelo· Compare with the current version
    Changes from Version 3

    […] is always a computer to which those "random" strings *aren't* random. So theThe extent to which patterns or randomness isare necessarily observer dependent is a genuine facet of math being limited. But further, there is a deeper point, which perhaps is what you are getting at, which is that it is not just the […] this experience? Maybe not. Some things are defined by their limitations. Can anything be defined from the outside? Can anything be defined from the inside? Not sure if I fully appreciate your post, but these are the sparks that were struck for me. Hopefully some are enjoyable.

  15. Version 3Henry D’Angelo· Compare with the current version
    Changes from Version 2

    […] complexity, which quantifies the randomness of strings relative to a computer. I'll use finite binary strings. Informally, the Kolmogorov complexity K(x) for a string x is equal to […] to find methods of compression that allow him to parsimoniously "recreate" (these datapoints, and in turn, predict, them (assuming his methods of compression identify true "hidden causes", and ignoring the problem of induction) these datapoints. Eventually the scientist may, inevitably, arrive on certain methods of compression. […] need numbers, and with enough time he is sure to discover them"something like" numbers. So perhaps, there are indeed methods of compression that are universal or "inevitable" for effective compressers with enough datapoints. This is what I imagine math aspires to. For example, maybe addition is one of those universal methods of compression. What I'm saying, then, here is that if you're trying to compress datapoints, and you have […] like" addition (up to isomorphism?) will be strictly useful to encode. Maybe. And, so any compresser that's good enough and has datapoints, will do so. […]

  16. Version 2Henry D’Angelo· Compare with the current version
    Changes from Version 1

    […] methods of compression that allow him to parsimoniously "recreate" (in turn, predict, assuming his methods of compression identify true "hidden causes", and ignoring the problem of induction) these datapoints. Eventually the scientist may, inevitably, arrive on certain methods […]

  17. Version 1Henry D’Angelo· Compare with the current version

    This is beautiful. I will respond to one specific thought --- the claimed target of math being omnipotence.

    To set the conceptual stage for my response, let me first appeal briefly to some notes.

    Question: How do you quantify randomness or "patternness"? One of my favorite proposals is the concept of Kolmogorov complexity, which quantifies the randomness of strings relative to a computer.

    Informally, the Kolmogorov complexity K(x) for a string x is equal to the length of the shortest computer program that can output x. So, what does this mean?

    Consider the ζ (which is just “10” repeated 50 times):

    ζ: 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010

    This string is 100 characters long. But its Kolmogorov complexity, K(ζ), seems to be less than 100. Why? Because you could (in a given language) program ζ in less than 100 characters. I essentially just did; in pseudocode, it would be something like:

    print(“10” * 50)

    This program, despite being shorter than ζ itself, would output the string correctly. So you get the idea: the Kolmogorov complexity of an object in effect reflects how long it takes to be described (though the specific measure varies across different languages and computers, the idea is consistent across).

    The basic thing to note is this: there is a sense in which ζ seems not to be random; intuitively, it is the fact that it has a pattern. And it is this pattern—repeating “10”—that enables us to describe ζ in a compressed way (that is, using a program that takes less characters than the string itself). Kolmogorov complexity is meant to formalize this intuition — that fundamentally, patterns enable compressibility, and randomness is incompressibility.

    So, Kolmogorov complexity (or algorithmic information) captures the intuition of patterns being dual to randomness, which is neat. That said, it's not generally computable, and there are some technical things needed to make it work formally (e.g. the idea of prefix-free Kolmogorov complexity).

    Now, how does algorithmic information relate to the question of math's potential for omnipotence?

    Informally, I generally think of math as follows: "the methods of compression, universal across effective compressers, in the limit of data."

    Let me make this more concrete. Imagine you are an "effective compresser". By which I mean, you are something that receives datapoints (sense-experience), and you can do some kind of "operation" to encode them. You also have time to do so. Example: a scientist who is reasonably intelligent. The scientist observes the world. He seeks to find a "theory" to understand these datapoints, and he has his lifetime to do so. His basic goal is to find methods of compression that allow him to parsimoniously "recreate" (in turn, predict) these datapoints.

    Eventually the scientist may, inevitably, arrive on certain methods of compression. Maybe he'll need numbers, and with enough time he is sure to discover them. So perhaps, there are methods of compression that are universal or "inevitable" for effective compressers with enough datapoints.

    For example, maybe addition is one of those universal methods of compression. What I'm saying, then, is that if you're trying to compress datapoints, and you have enough datapoints, eventually "something like" addition (up to isomorphism?) will be strictly useful to encode. And so any compresser that's good enough and has datapoints, will do so. That is to say, perhaps any scientist that is sufficiently advanced and has enough time / data, will eventually discover addition.

    Quantity, length, etc. these things strike me as being potential "universals" in this above sense.

    But then...

    Are there such universals? Does this concept even make sense?

    A problem is that Kolmogorov complexity is computer/language dependent. You can force it to be a universal computer, sure. But the reality is that relative to any computer, even a universal one, the vast majority of strings are random (I can prove this fact quite easily, if you'd like). But then there is always a computer to which those "random" strings aren't random.

    So the extent to which patterns or randomness is necessarily observer dependent is a genuine facet of math being limited. But further, there is a deeper point, which perhaps you are getting at, which is that it is not just the question of whether these "datapoints" (left undefined) are patterned (compressible) or not, BUT the fact that it is a kind of fluid act or "process" of actively receiving datapoints, being affected by them, making sense of them, etc. that is then itself a totally different thing. You don't get anything whatsoever about firsthand experience from "universals"; in fact, firsthand experience might be described as "that which is not universal".

    A scientist who observes the world and discovers some beautiful pattern or order; this is a beautiful thing. Is there any way to "compress" this experience? Maybe not. Some things are defined by their limitations.

    Not sure if I fully appreciate your post, but these are the sparks that were struck for me. Hopefully some are enjoyable.