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There are definitely states of systems(like mind) which are not quantifiable. For mathematics to work in principle, we need states which are quantifiable or measurable. So, does this go to show that complete description of reality in mathematical terms not possible?

David Chalmers argue the nature of consciousness, which is responsible for subjective experience, is something innate to the universe. An example he often cites is Mary a neuroscientist who knows everything, that is to know physically, about the colour red will still not know colour red when she first experiences it.

Also, Wittgenstein in Tractatus argues that

A logical picture of facts is a thought.
A thought is a proposition with a sense.

But it is generally agreed upon that this leaves a lot which can be claimed non-sense in Wittgenstein. As he himself acknowledges in his last proposition

Whereof one cannot speak, one must be silent.

So, if there are states of the world which cannot be appropriately expressed even in language, how can mathematics describe such states?

So, a further question is whether reality logical?

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  • Please leave a comment as to why you think this to be an inappropriate question. I have for sometime have this question - how far does reality can be described by mathematics?
    – prateek
    Feb 7 at 16:09
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    Who knows? And why assert that "there are states of the world which cannot be appropriately expressed even in language". If we cannot describe them with language, what/where are them? And mathematics is language. Feb 7 at 16:49
  • By language there I meant the language of common discourse. Yes, mathematics is a language but it is not as flexible as, say, English. So, I think there could be propositions which can be expressed and understood in English but not in mathematics - for example, say - Coffee is 'very' hot. Yes, I understand that it is a question nobody has a perfect grasp on but still that doesn't mean that some people genuinely engage with it. So, my purpose in asking this is for someone to point to current state of research/thought about this.
    – prateek
    Feb 7 at 17:05
  • Also, we know that Physicists casually talk about a theory of everything. Will it not be helpful, if someone can clarify on what physicists mean by everything. Anybody after such a theory will surely have to understand what their theory will or will not answer. So, that also assumes answering this question apriori.
    – prateek
    Feb 7 at 17:09
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    There are plenty of objects in mathematics that cannot be expressed in English or in mathematics. For example, almost all real numbers. The reason is simple: the set of English sentences or mathematical formulas is countable. The set of real numbers is larger than that - uncountable. So the vast majority of real numbers can't be named by a formula or an English sentence.
    – causative
    Feb 7 at 19:40
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Completeness in mathematics has a specific meaning. Godel's Incompleteness Theorems showed this is not possible, for mathematics as a whole, and ended most of the remaining parts of the Hilbert Programme, including the aim to axiomatise physics.

Stephen Hawking grappled with the consequences for physics here, and the nature of what a Theory Of Everything would be: Godel and the End of Physics.

Godel's theorems are anti-foundationalist - a 'final vocabulary' is not possible. This is because minds, who create and use language, are strange loops, with tangled hierarchies, that include self-references and feedback loops. For a mind to understand the world, it must also understand itself, which complicates itself, requiring more understanding, a task which can never be completed. Minds are dynamic, creative, and exist as interactions, including through intersubjectivity. The best possible understanding must also be dynamic, interacting, alive.

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    As a minor quibble with the opening, completeness in mathematics has multiple specific meanings - and Godel himself also showed that in a different sense, the standard logical framework for mathematics is complete! See here. Feb 7 at 19:20
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    As a more serious objection, all Godel's incompleteness theorem rules out is a computably axiomatizable complete consistent theory satisfying a certain technical "strength" condition. So there's a fair amount of nuance here - a complete theory is possible, it just has to be rather complicated. Feb 7 at 19:21
  • @CriglCragl here are you assuming there to be a one to one correspondence between mathematics and reality? For as I understand even Hawkings is saying that Godel's theorem is to mathematics as M-theory is to physics. He is only implying that as Godel's theorem holds for mathematics there could be a theory in Physics which is incomplete but that is not a foregone conclusion because we are still pretty far away from a physical theory which describes all the phenomenon we have catalogued...
    – prateek
    Feb 8 at 12:09
  • ...So, in principle I think there is scope for laws which apply to all of physical world like the 2nd law of thermodynamics which results in newer and newer exotic objects with very different physics emerge from more basic components.
    – prateek
    Feb 8 at 12:10
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    @CriglCragl - A version of Godel's theorem might say that no set of laws can tell us whether an arbitrary physical computer will ever halt or not (and it might have implications for other types of physical processes that can potentially go on forever but can also potentially stop depending on initial conditions). But Godel's theorem gives no reason to think we couldn't find the exact laws that determine what state an arbitrary system will be in at time T1 given sufficient knowledge of its state at time T0, where T1-T0 is some finite time interval.
    – Hypnosifl
    Feb 11 at 4:55
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To be able to answer "yes" would require a complete definition of "reality". The more we learn about our universe...our reality...the more we realize how much we don't know. Lacking that complete definition, this question is unanswerable.

A question possibly worth asking is "Will our understanding of mathematics ever be sufficient to fully describe some notion of ours of what reality might be?". To that question, looking just at what's been said in the answers to this question, I'd say...not likely.

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It depends on the laws of physics.

First, note that there are plenty of objects in mathematics that cannot be expressed in mathematics. For example, only a tiny fraction of real numbers are named by any mathematical formula. The reason is simple: the set of mathematical formulas is countable. The set of real numbers is larger than that - uncountable. So the vast majority of real numbers can't be named by a formula.

However, it is possible to describe certain laws governing all real numbers, even though most of them can't be individually named by a formula.

Now, here are some possibilities for the laws of physics, together with the consequences of each possibility for how well mathematics can describe the laws of physics. Each possibility may fall into one of the following categories: computable, computable in the limit (approximable), not computable but describable, not computable or describable.

  1. It may be that fundamentally, space and time are discrete. More than that, it may be that the evolution rule that derives the next state of the universe from the previous one, happens to be computable. If this happens to be true, then mathematics can not only describe everything in the universe - given enough information, a Turing machine could perfectly predict all of it. (category = computable)

  2. It may be that fundamentally, space and time are discrete, but the evolution rule is not computable. In this case, no matter how much information we gather, no computer will be able to perfectly predict what the laws of physics will do. However, it may still be possible to specify what the laws of physics are in the abstract, even though we can't calculate them. (category = computable in the limit, or category = not computable but describable)

  3. It may be that fundamentally, space and time are continuous, but the laws of physics can be described by differential equations we can write down. For example, the motion of a pendulum is continuous but described by a simple set of differential equations. If this is the case, mathematics is able to describe the laws of the universe. Depending on the specific laws, it may or may not be possible to progressively approximate the result of a physical process as closely as we wish. (category = computable in the limit, or category = not computable but describable)

  4. It may be that fundamentally, space and time are continuous, and the laws of physics are not describable in mathematics. This is conceivable because the set of all possible laws of physics is very large - at least as large as the real numbers, because a law of physics may include a real number as a parameter. And the set of all possible laws of physics that can be named by a mathematical formula is much smaller - countable. In this case, no mathematical calculation can perfectly approximate or even describe what's going on in the universe. (category = not computable or describable).

All four of these possibilities are logically possible. Philosophy ultimately cannot say whether the universe can be described by mathematics; that is up to the physicists. It's conceivable that physicists might find a Theory of Everything that perfectly describes the universe. It's also conceivable that the laws of the universe are fundamentally not describable by mathematical formulas. It's up to physicists to find out.

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The term "reality" is far too general to make much sense of this. But I believe that the answer is clearly Nein! Any mathematical description of "reality" would be part of that reality and must therefore describe itself, leading to the paradoxes of self-reference in set theory and to a Droste-like infinite regress.

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    Imagine (counterfactually) that the laws of reality happened to be Conway's game of life. Conway's game of life is Turing-complete. It is possible, therefore, for a pattern in Conway's game of life to emulate a Turing machine that itself is running Conway's game of life; there is no problem with self-description in this sense. Tangentially, check out the concept of a quine: en.wikipedia.org/wiki/Quine_%28computing%29 . There are many legitimate ways for systems to describe themselves without paradox or contradictions. Only some ways are problematic.
    – causative
    Feb 7 at 22:37
  • In case it wasn't clear, a Turing machine emulated within Conway's game of life, that is itself simulating Conway's game of life, is an exact metaphor for a computer within our universe that itself is simulating our universe. Obviously this Turing machine cannot simulate the entire grid in which it sits, but there is no problem with having the Turing machine simulate a smaller pattern than itself, which is what computer simulations do.
    – causative
    Feb 7 at 22:40
  • Interesting. I don't know enough math, but mathematical self-replicators always seemed fishy to me, especially in the presumably physical, frictional, thermodynamic context of "all of reality." And wasn't the Turing machine itself originally a demonstration of incompleteness? I just get stuck on any meaningful idea of a "complete self-description." Feb 7 at 23:05

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