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To what extent mathematics can capture all physical phenomena? Drawing an analogy from computer science: finite automata can handle regular expressions (does "(([a-z]))" match "((h))" ), but for more complex tasks, we need Turing machines (eg. determining if a string has balanced parentheses). There is no way we can physically "blow up" finite automata to where they can solve turing computability problems. In the same vein, is there a limit to where we can "blow up" math to where it won't be able to solve the next complexity of "physical phenomena". Where on the computability hierarchy do physics even falls under.

These limitations could have implications on the current open questions on unifying gravity with qm and other theories of everything suggesting them to be fool's errands.

I've already read this thread (Why is mathematics so fantastically successful at describing the universe?), this is a kind of asking the opposite to see where we stand.

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    I started to answer your question, but quickly concluded that I was having to guess what you meant by 'fully describe'. Can you clarify what you mean, please? Oct 22, 2023 at 6:21
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    The analogy is too limited to draw any grand implications. There is a big difference between describing and computing, we can easily describe theories whose theorems we cannot compute. And even if we cannot describe a description may still exist. Many physicists believe in a mathematical "theory of everything" that covers all the laws governing the universe. Even if we were to discribe it we still would not be able to derive all of its theorems or make all predictions with those we do derive (initial conditions are also needed). Quantum gravity is a much more modest task, and likely doable.
    – Conifold
    Oct 22, 2023 at 7:59
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    In my opinion the answer to this question is, briefly, this:- As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. - Albert Einstein (When he delivered his lecture “Geometry and Experience” at the Prussian Academy of Science in Berlin on 27 January 1921). People forget or ignore the awareness that the distinction between reality and non-reality is a distinction arising only in the human mind. Reality (physical universe) is whatever it is independent of the concepts arising in the human mind. Oct 22, 2023 at 20:34
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    it's not even clear Math can capture any physical phenomenon at all. The map may not be the territory. Even if it is, it's not obvious that math can explain what it feels like to experience something, rather than just explaining what something is. Oct 22, 2023 at 21:23
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    Surely if it can't we will simply change what "mathematics" means to include the new stuff we need. Oct 23, 2023 at 20:11

15 Answers 15

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No, by Godel's incompleteness theorem. Since no formal axiomatic system can describe all truths of arithmetic, and arithmetic is necessary to physics, physics can't fully describe the universe.

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  • This is what actually prompted the question. There are some other good answers/comments such as the one where it said math isn't enough to file taxes, why would it be enough to describe all of physics but they all boil down to this theorem/idea. Marking as the technical answer to my question but hope to live long enough to see the question get answered in my lifetime.
    – PHV
    Oct 25, 2023 at 22:10
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    This is a bit wrong: Gödel's incompleteness theorem says this but perhaps, we don't need all truths of arithmetic to describe the universe. You are assuming that "Physics uses arithmetic" and hence, "it must use all of arithmetic, even those "parts" affected by GIT." Perhaps there is no guarantee to that.
    – Red Banana
    Oct 28, 2023 at 14:28
  • Except that it's absolutely not what Gödel's theorem of incompleteness says.
    – Evpok
    Feb 3 at 23:38
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Clearly, no mathematics is ever going to describe redness and pain and love to any reasonable person's satisfaction.

To describe something is to say what you think this something is by using a subject-predicate structure. You use the sentence "the glass is full" to describe the glass as being full. This can go a long way, as demonstrated by science, and in particular quantum mechanics and general relativity, but at some point it can go no further. What description would convey what redness is? The statements "Redness is red" or "Redness is redness" would be uninformative and therefore no descriptions.

Mathematics is nothing if not the use of the mathematical language, which is itself just the part of natural language involving a specialised terminology, symbols and a restricted grammar. The logic of mathematical proof is exactly the same as the logic of natural language arguments. What a natural language cannot in principle do, the mathematical langage cannot either.

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Mathematics can be used to make a model of the universe. All models are necessarily simplifications of the thing they model - if they weren't they would be of no use as they would be no easier to understand. Hence the famous quote from G.E.P Box:

All models are wrong, but some are useful.

Models never fully explain anything, they always ignore some things, such as initial conditions of chaotic systems that cannot be known with complete accuracy, or systems comprising of very large numbers of variable (such as applications of statistical physics) where we instead have to focus on statistical aspects (the laws of thermodynamics are essentially statistical "laws" not physical ones).

Mathematics can't fully explain the universe, there simply isn't enough room for the universe and the encoding of the explanation, especially if they are both infinite.

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"Can Mathematics Fully Describe the Universe?"

Certainly not, even with the current state of affairs. Physicists use many entities that currently make no (strictly) mathematical sense, such as the Feynman integral. While we can hope that things may change in the future, currently mathematics can't even fully describe the currently existing physical theories of the universe.

As noted in the comments, physicists recognize that only certain Feynman integrals are mathematically well defined; for most cases they only use them as a recipe for perturbation theory and don't know the correct mathematical formulation for the "full" object (or if it even exists). Hall in his "Quantum Theory for Mathematicians" writes:

"To have a chance to make rigorous sense of path integrals in quantum field theory, one has to employ a complicated regularization process known as renormalization. This process has, so far, been carried out in a rigorous fashion only for a very small number of field theories. One of the Clay Millennium Prize problems is to make rigorous sense out of the Yang-Mills field theory in four spacetime dimensions." (pages 451-452).

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The title question could be better addressed to physics than to mathematics.

Because it is not the primary goal of mathematics to describe the universe. The claim of mathematics is much more modest. Mathematics provides precise concepts and a formal theory, which can be used by physicists and some other scientists to formulate questions and answers from their specific domain.

The reasearch on string theory during the last decades shows, how progress in the field of mathematics is stimulated by the request from the physicist for help to formulate their physical ideas. And to answer the mathematical questions arising from this formalization.

Your question about a hierarchy of different levels of mathematical power is interesting. I would be interested whether your question can be stated in a more formal way than your analogy with the Turing hierarchy.

I do not know and I will not speculate whether mathematics will be able to satisfy also in the future the needs of physics in all its aspects of exploring and explaining the universe.

PS. To close with a quote ascribed to Einstein :-)

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

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    "If all mathematics disappeared, it would set physics back precisely one week." - Richard Feynman. Mark Kac responded: "Precisely the week in which God created the world."
    – David S
    Oct 24, 2023 at 17:04
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If the Universe is finite, then yes, because everything and the properties of everything can theoretically be enumerated. If the Universe is infinite, then no, because of Godel's Incompleteness Theorem. Unless, of course, one takes a broader view of the term "Mathematics" than the proving of statements in formal systems. In which case, I suppose you can define "Mathematics" in a way that either is or is not capable of describing the Universe as you see fit.

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  • Incidentally, Godel's theorems are basically corollaries of Turing's results and vice versa. Both problems essentially hinge on infinity (as in the natural numbers). Whatever issues you find with our ability to undertand the Universe from a computational point of view, you shouldn't expect to gain anything by pivoting to Mathematics instead.
    – Him
    Oct 22, 2023 at 13:43
  • I’m nowhere near so sure. I’m not even comfortable that the question is meaningful. OP asks about “all physical phenomena.” It seems to me that there are infinitely—perhaps uncountably—many of those, even for a finite system. I suspect that my hang-up about OP’s all is at least closely related to others’ objection to describe. And I guess further that it boils down to What constitutes a physical phenomenon. Oct 22, 2023 at 17:46
  • When you say that the universe is "finite", what do you mean exactly? Do you mean the amount of particles? Is the amount of configurations of said particles also finite? Oct 22, 2023 at 22:06
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    Eventually you run out of time, so the number of configurations has to be finite. If you don't run out of time, you run out of energy gradients to cause change.
    – Scott Rowe
    Oct 22, 2023 at 22:16
  • @MartinArgerami I think the primary requisites for some kind of meaningful "finiteness" would be finite extent and finite divisibility. As in the universe has an "edge" or alternatively wraps back onto itself (it's a torus, or some such), and also there is a limit to "zooming in". Quantum mechanics seems to suggest that this might be the case, but it's not entirely clear that small scales don't merely "appear" discrete and are "actually" continuous or even simply countably infinite (e.g. allowing only rational divisions)
    – Him
    Oct 22, 2023 at 22:46
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I suppose the question is equivalent to asking if there is such a thing as quality, of a type that would be irreducible to quantity. Because of course, what is quantifiable will get quantified, what can be measured will be measured, and so forth, but at the end of times when we have quantified everything quantifiable, will there be a residue? Something left unquantifiable? Something non-analysable, ineffable?

Like what? The feeling(s) of being alive, the wind on one's face, the taste of oranges, the love for that someone - existence? Maybe. Even if arguably, qualia such as tastes and smell are genetically coded and biologically produced, and even if the genetic code is some kind of chemical math, feelings and emotions ought not be reducible to math... There are like the substance of our existence, and of our apprehension/apperception of nature and ourselves, respectively. Both qualia and emotions have, to us, a qualitative nature or appearance that seem impossible to communicate by words, and hence even more impossible to put in numbers.

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  • This restricts the domain of mathematics to what is "quantifiable", which im not sure is unreasonable, but its hard to see now certain mathematical ideas have anything to do with real world quantifiability. Infinity is an extremely important concept in maths, and its not clear how it is quantifiable in any real world sense. Similarly for any irrational number. Oct 23, 2023 at 2:42
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    @MichaelCarey Irrational numbers are evidently quantifiable, as numbers.
    – Olivier5
    Oct 23, 2023 at 6:19
  • thats'a certainly true, but everything in mathematics is "quantifiable" in some sense inside Mathematics. I think the common parlance of quantifiable is "quantifiable data", I can't really see us ever being infinitely precise, and being able to express some actual object as being an irrational number. Ofc, if the poster meant something more precise/differentby quantifiable- then I'm with them. Even if not, it's not that strange of a restriction of math. Oct 23, 2023 at 9:17
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    Combinatorics includes "bean counting", and counting itself is essential to mathematics- which is subsumed in the study of cardinality. But, I dont see how say, Godel's Incompleteness Theorems could ever be reduced to "counting" in any meaningful way. And, I don't really think stretching quantity to include things like Topology or Differential Geoemetry is a useful way to think about those topics. Personally, I think of Mathematics as the study of formal abstraction. Where "formal" relates to being expressable in the language of Mathematics. So something like a first order language. Oct 23, 2023 at 11:30
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    @MichaelCarey OK so you are saying that "foundational math" are pre-quantitative, in the sense that set theory for instance helps define numbers.
    – Olivier5
    Oct 26, 2023 at 10:19
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Phrasing the dissents a different way, one must define what it means to "fully describe" a phenomena. It turns out "describe" is a trickier word than one might think. One can describe all the phenomena in the universe as "all the phenomena in the universe," but such quippy answers are clearly unsatisfactory. What proves tricky is excluding such quippy responses while still leaving something mathematics has a chance of describing.

I had a category theory teacher who once said he wasn't sure if functional programming was the language of the universe, or if it were simply the limit of the way the human mind thought about the universe -- and was inclined to believe it was the latter. If that is the case, then the impact of a statement like "Mathematics fully describes the universe" is less a statement about the universe, and more a statement about our minds.

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  • Math fully describes the universe... for our limited ability to comprehend. The hammer is heavy enough to drive nails we use, and not so heavy that we can't lift it. Although most math is beyond me.
    – Scott Rowe
    Oct 24, 2023 at 10:41
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There is the idea that the universe actually is a computer which immediately answers your question.1

I suppose that your intended question could be rephrased as whether it is possible to perfectly and completely simulate this computer with a smaller computing scheme (math/physics), to which the answer is no, both because the universe is its best simulation and because you'd run into logical impossibilities akin to the Halting Problem: You'd have to recursively simulate your own simulation, which is part of the simulated universe, after all.


1 A word of caution. Each age has a dominating paradigm which informs the concepts and metaphors it uses to get a mental handle on its surroundings. When myths and magic and ancestors dominated our predecessors' thoughts, the universe was believed to run on those; when mechanical clocks were developed, the universe was thought to be a giant intricate mechanism. With the slightest bit of detachment it is not astonishing that we might have the idea that the universe be a giant computing machine. Just sayin'.

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  • The simulation hypothesis is probably the field of study of this question. Oct 25, 2023 at 11:43
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Mathematics works over abstractions of nature. Three apples in a bag can be described as this:

b = 3a

However, mathematical abstractions need of a rational mind to have a correspondence to the world. Without someone's mind, the same formula can be understood as "three ants are equivalent to one bee".

So, in order to describe the universe, you need:

  1. A mind which is able TO KNOW the whole universe. That is, from ALL and each one of all possible quarks to the whole set of galaxies. Evidently, this is exceeds our capabilities by infinite infinities.
  2. A formula which would describe the whole statics (all objects) and dynamics (all interactions) of the universe. Including itself. Evidently, this is exceeds our capabilities of writing formulae by infinite infinities.
  3. An act of understanding: the mind in (1) should be able to read the formulae in (2), and understand them. Evidently, this is exceeds our capabilities of understanding formulae by infinite infinities.
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    You seem to be arguing that since human mind is finite, it cannot fully comprehend infinite entities. Would you apply this reasoning to the natural numbers N as well? Oct 23, 2023 at 12:04
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    @MikhailKatz Metaphysical entities -as the natural numbers- are knowable. Here we are talking not about metaphysical facts, but about physical facts, which are strictly not knowable (see noumenon).
    – RodolfoAP
    Oct 23, 2023 at 17:30
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I see many answers full of certainties, with quotes from Godel's theorem and other sentences that seem to demonstrate that mathematics is a representation of reality and as such cannot explain all of reality. The truth is that the answer is neither certain nor already determined. For example, Max Tegmark, in his book Our mathematical universe, supports the risky hypothesis that mathematics is the only true reality and what we see is only a partial representation of mathematical constructs. Personally I don't have a clear opinion on the subject, however I think Tegmark's book is quite convincing in explaining how he reached that conclusion. Furthermore, the hypothesis that the universe is just a big quantum computing system, supported by Fredkin and also explained in the Lloyd's book Programming the universe is a very interesting theory and taken into serious consideration in the field of physics.

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No, never; not even given all eternity in which to try, for two very different reasons.

Far and away less important is that any outcome would depend not on maths but on the ability of the best mathematician.

Less obviously and much more importantly consider the tiniest thing you can imagine, and try to describe it.

Does it not have at least height and width, depth and mass if not many another important characteristic?

How could any language, including maths, begin to list, let alone 'fully describe' anything about that tiniest thing in any way even equal to, let alone 'smaller' than the thing itself?

For a basic analogy, how much ink and paper might your pen need to 'fully describe' a full stop and how could the volume, mass or any other characteristic of ink or paper, let alone both, be 'smaller' than the thing they tried to describe?

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To the extent that we can model these physical phenomena as mathematical constructs. There is no theoritical limit to it, althought there is a practical one: the interrelations of these phenomena produce states in physical reality that cannot be accurately modeled due to complexity; but there can be approximations.

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Max Tegmark says that the Universe is Mathematics. I happen to agree with his insanity, at least in part.

There is mathematics, and then there is Mathematics. The mathematics we humans know about are systems represented by symbols that we understand. The Mathematics that is the Universe is, for the most part, way beyond the understanding of mere mortals, at least for the moment.

So yes, Mathematics fully describes the Universe because Mathematics is the Universe. We humans have scraped out, in our mathematics, just a bit of the Mathematics.

Will mankind be able to fully embrace Mathematics someday? Who knows?

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No it cannot not. The reson for this is that mathematics is not the most fundamental science. What was mentioned even in Euclid? Yes, systems! Theory of systems is the most fundamental science.

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