Is the Universe Made of Math?
scientificamerican.com
scientificamerican.com
Cue to Whitehead's "a series of footnotes to Plato" quip. Though I think he was a bit off the mark. People aren't merely talking about Plato, but unwittingly echoing him. It's kind of like that another quote, this time from Keynes, about practical men being "slaves" to the opinions of long dead economists.
EDIT: A quick read of the relevant Wikipedia entry shows that Tegmark's view is basically good-old mathematical Platonism, with a radical, monistic twist.
EDIT2: A question I would pose to a mathematical Platonist monist like Tegmark would then be, what is the cardinality of the collection of universes? Brouwer could reject large infinities only because he insisted that mathematics is limited by the physical, but if the physical and the mathematical are the same, I can see no justification for rejecting a large set of universes.
[1]: https://plato.stanford.edu/entries/philosophy-mathematics/
[2]: https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...
That ignores the issue of how useful this description is. It is easy to conceive of an external mathematical reality that cannot be fully described from within. Even with a full description, there is no guarantee the description is at all useful. If you consider how stumped we are by the goldbach conjecture, and how simple that is, you realize there are plenty of mathematical structures that defy understanding.
And then there are the standard crises of maths. Incomputable problems, paradoxes in informal systems, controversial axioms in formal systems (axiom of choice), and that final bombshell of Godel's incompleteness theorem. All of these make math seem less rock-solid. Heck, if the universe is maths, we can't prove the universe is consistent from within the universe.
What a lot of people do not see, including many theoretical physicists is that mathematics is a "map". It describes reality in a simplistic way, but is not the reality itself.
twlevechairs26 refers to the quote "All models are wrong but some are useful". This highlights the fact that all models are maps, incomplete, simplified descriptions that leave out specific parts or include things that are not part of the thing being described.
When we get arrogant enough to claim the "truth" in mathematics or science we have moved into the realm of religion. Mathematics and science helps us get a handle on trying to understand the physical universe around us. But it will NEVER, and I repeat this, NEVER give us perfect understanding of the universe around us. For that perfect understanding would require that we become greater than the universe around (in effect raising ourselves by our bootstraps to be God or gods).
So, in may ways, the first question here, "Is the map the territory or not?" before asking "Is the Universe made of mathematics?"
How can you be sure?
>or that perfect understanding would require that we become greater than the universe around (in effect raising ourselves by our bootstraps to be God or gods).
shrug
[1] https://www.forbes.com/sites/janetwburns/2016/10/13/elon-mus...
It's astounding that someone as obviously intelligent as Musk believes in creationism.
Both statements would be true.
IMO, Tegmark has a solid argument that the MUH is more plausible than any of the other suggested possibilities (https://www.amazon.com/Why-Does-World-Exist-Existential/dp/0... is good for the PEQ and much more worth reading than Tegmark's book).
Physics may be mathematical, but that doesn't mean you can start from the Big Bang and use mathematical physics to predict the contents of this particular thread on Hacker News.
So you immediately have the problem that mathematical physics cannot possibly be a complete description of the universe.
Of course you can argue that we simply don't know the math yet, and if we did we'd be able to start from the primordial singularity and predict the existence of Max Tegmark, Y Combinator, and me and you.
But that's a very strong claim, and I'm not aware of any evidence that comes close to supporting it.
Even if you make a much looser claim - that mathematical physics can predict classes of phenomena like Max Tegmark, etc - there's still no math around at the moment that even begins to pretend it can formally prove that assertion.
Worse - strong claims about mathematical provability immediately fall apart because of Godel. How can you have an abstract formal system running the universe if that system can't possibly be complete, provable, and formally consistent in its own terms?
incompleteness means true but unprovable propositions in specific forms of higher order logic, but Kurt also proved the completeness of first order logic. However, the scheme of induction used to lower peano arithmetic to first order logic (Presburger Arithmetic e.g.). Likewise, an incomplete higher order logic can (ostensibly - I couldn't) be proved in a logic of even higher degree, which is again incomplete, so again that's an argument ad infinitum. Obviously that's abstract nonsense, but the question shouldn't be whether the universe can be predicted up to infinity. The question at face value is rather, whether quantitative analysis can solve any given qualitative problem. It's the syntax semantics duality - what comes first?
The incompleteness theorems hold for any formal system whatsoever, regardless of order. They are simple corrolaries of the halting theorem. First order logic is indeed complete in the following sense: A sentence of FOL is provable iff it's valid, i.e. true in all interpretations. But according to the incompleteness theorem, logical theories (sets of axioms) of a certain richness would always yield sentences that are true in some interpretations and false in others, or, alternatively, that are neither provable nor disprovable (and first-order Pano arithmetics and obviously all formal set theories are among them).
Order and other particulars of a formal system are irrelevant, and indeed Gödel was only fully satisfied with his incompleteness theorems after he read Turing's famous paper on computability, and regarded it a "miracle" (as it applies to any formal system, and depends on none).
Obviously, many quantitative problems are unsolvable as a direct result of the halting theorem, and more generally the time-hierarchy theorem (which states that there are problems of any arbitrary difficulty), as many decidable problems are still unsolvable due to intractibility.
However, there is the calculus of constructions, which, as far as I can tell, tries to be provable in all admissible formulas. It's a totally functional language. So, is the universe totally functional, will it halt? I'm not sure that's the actual question, neither whether the universe is beyond our understanding, that's out of the question, only, does the universe follow rules or is it deterministic and not only in the quantum sense.
> all computable mathematical structures (in Gödel's sense) exist - en.wikipedia/wiki/Mathematical_universe_hypothesis
That's trivially true, because they exist in the heads of the mathematicians constructing them.
Total programming languages (or more precisely, their term language), is not Turing complete, and is not universal (although such languages, like Agda, are Turing complete, but not in their terms; i.e. their expressions are not Turing complete, but they allow simulating Turing machines using an interaction with the runtime using something called codata). If the universe is finite, then it, too, cannot fully simulate a Turing machine (and the formalism, or language, used to desrcibe it is completely irrelevant; total functional languages aren't just formalisms, and the universe is a true system).
> That's trivially true, because they exist in the heads of the mathematicians constructing them.
Well, it's not so simple. First, we have to philosophically determine what we mean by "exists". Platonists believe that there exists a world of abstract forms. If we restrict existence to physical existence, we still need to determine what it means for an abstract mathematical structure to exist in the mathematician's mind. Brouwer believed that for those structures to "exist", the mathematicians should be able, at least in principle, to construct them. So only constructive math "exists". On the other hand, Hilbert, also a finitist, said that math is made of two parts: "real" propositions (finite, about the physical universe), and ideal propositions (finite expressions but infinitary objects, those don't "exist" as anything other than the finite formulas that express them). But even in Hilbert's formulation, there are many inifinitary mathematical objects for which no finite description could exist (by a simple cardinality argument). Do those exist? For Hilbert, probably not, but for Platonists, they still do.
Only because the only available approaches for even beginning to do anything of that sort are constrained by being embedded in that same universe.
It's unsurprising that a system cannot exist, and at the same time embed a calculation which calculates all of itself.
(We cannot even use mathematics to predict simple N-body problems; we have to switch to numerical techniques which have limited precision. Once you have floating-point in there, it's no longer the math, but an approximation.)
Equivalence is a weird concept though. Is the factor group O(3)/SO(3) equivalent to Z/2Z? As a group, the answer is yes. As transformations of R^3, most definitely not.
- If you have no access to other universes, how can you design an experiment which positively confirms their existence?
- The situation without hidden variables is indistinguishable experimentally with a situation that does have hidden variables.
The latter is precisely why we can make a virtual machine which makes the kernel believe that it's running in a PC chassis with an Intel EEPro 100 ethernet card.
In fact, based on Tegmark's hypothesis, I have to suspect that even the universe I'm apparently in doesn't just exist stand-alone; it's also embedded/contained an infinite number of times in more complex universes (which virtualize so perfectly, that there is no way to tell). One of those universes is the simplest one, not embedded in anything; for all intents and purposes, we can proceed as if we are in that one.
I've read that the author may have some opinions regarding AI that may be considered religious, but this particular subject so far seems purely philosophical to me.
I consider it perfectly justifiable to call it physics, but at this level of abstraction, it's hard to differentiate it from religion too.
I think that the 'of course' here is not at all obvious. Economics, for example, seeks to describe how people will behave, but neither makes nor, I think, is perceived to offer any description of how they should behave (at least not in any moral sense).
Anyway, even a deterministic universe (mathematical, i.e. monistic Platonistic, or otherwise) is still undecidable/intractable, and so normative prescriptions may well serve an important role as you don't know their place in the causal chain.
Of course, you could argue that "importance" is also a subjective measure in a deterministic universe, and an irrelevant one, but then so is "relevance" itself.
As long as he doesn't say how people should behave, this isn't religion (even if it relates to how people will behave).
[1]: I.e., assuming a distinction between physical and mathematical
Plus, the faithful themselves were very skeptical of them, and lots of people were charged with impiety for expounding this kind of stuff. Specially so in the case of Spinoza and Kant.
But there is a difference between a universe in which math exists, and a universe that is made of math; and the latter is, I think, a much stronger requirement. (Unless, indeed, every conceivable universe is made of math, in which case the requirements have exactly the same strength. :-) )
> Copy rights
is that a pune?
Given that math can easily describe conditions that aren't realised in any obvious way in our universe, this seems like a proposition for whose meaningfulness it would be hard to argue. (Granted, the math we do is realised in the universe, but of course "the map is not the terrain".)