# Are all models wrong?

George E. P. Box is famous for the quote: “all models are wrong, but some are useful” (Box, 1979). A statement that many modelers swear by, often for the wrong reasons — usually they want preserve their pet models beyond the point of usefulness. It is also a statement that some popular conceptions of science have taken as foundational, an unfortunate choice given that the statement — like most unqualified universal statements — is blatantly false. Even when the statement is properly contextualized, it is often true for trivial reasons. I think a lot of the confusion around Box’s quote comes from the misconception that there is only one type of modeling or that all mathematical modelers aspire to the same ends. However, there are (at least) three different types of mathematical models.

In my experience, most models outside of physics are heuristic models. The models are designed as caricatures of reality, and built to be wrong while emphasizing or communicating some interesting point. Nobody intends these models to be better and better approximations of reality, but a toolbox of ideas. Although sometimes people fall for their favorite heuristic models, and start to talk about them as if they are reflecting reality, I think this is usually just a short lived egomania. As such, pointing out that these models are wrong is an obvious statement: nobody intended them to be not wrong. Usually, when somebody actually calls such a model “wrong” they actually mean “it does not properly highlight the point it intended to” or “the point it is highlighting is not of interest to reality”. As such, if somebody says that your heuristic model is wrong, they usually mean that it’s not useful and Box’s defense is of no help.

On the opposite end of the spectrum are abstractions, these sort of models are rigorous mathematical statements about specific types of structures. These models are right and true of their subjects in any reasonable definition of the words. They are as right or true as the statement that there are infinite number of primes; or that in Euclidean geometry, the tree angles of a triangle sum to two right angles. When somebody says that an abstraction is wrong, they mean one of two things:

1. It is mathematically false. For example: if I say that given the first n primes $p_1,...,p_n$, their product plus one ($N = p_1p_2...p_n + 1$) is prime then that statement is false. It was a statement that was no made carefully enough (N is not divisible by any of $p_1,...,p_n$, but could be a composition of other primes not on your list).
2. Or, the structure you are applying it to does not meet the requirements of the abstraction. For example, in general relativity, space is non-Euclidean, so triangles don’t sum to 180 degrees. More concretely, if you are measuring triangles on the surface of the Earth then at large scales you can have triangles that sum to more than 180 degrees, even if on local scales Euclidean geometry is a good approximation.

In the first case, your model is not actually an abstraction since it is not valid on all structures meeting its prerequisites (but maybe you can convert it to a heuristic). In the second case, your critic is actually saying that your model is not useful, and Box’s defense is again of no help.

This brings us to models in physics and, more generally, to insilications where all of the unknown or system dependent parameters are related to things we can measure, and the model is then used to compute dynamics, and predict the future value of these parameters. Now, if we look at any currently extant model then just based on our history of model improvements, it seems presumptuous to assume that this model is not wrong. As David Deutsch eloquently wrote at the end of the Beginning of Infinity:

[S]cience would be better understood if we called theories “misconceptions” from the outset, instead of only after we have discovered their successors. Thus we could say that Einstein’s Misconception of Gravity was an improvement on Newton’s Misconception, which was an improvement on Kepler’s.

However, the question remains: is it conceivable that there is a perfect infallible model of the universe? Is the world fundamentally mathematical and comprehensible? As long as I can remember, I believed that the world is not inherently mathematical, that mathematics is a reflection of our conscious and there is no reason to expect that it can perfectly capture external reality in any sense of the word. In fact, I have always asserted that assuming the opposite is anthropically arrogant.

I think that many people agree with this sentiment, but some do so for the wrong reason. This wrong reason is usually a reiteration of On Exactitude in Science — Jorge Luis Borges meditation on the map-territory relationship:

In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast Map was Useless

Loosely, the argument is that a perfect model of the universe has to incorporate every detail of the universe and thus must necessarily be the universe itself. I don’t think this is a valid argument because the map analogy for insilications is flawed, and there might even be a deep reason for this flaw. In particular, if we concentrate on Tarski’s elementary geometry then our theory is consistent, complete, and decidable and thus not capable of any deep self-reference. If we lived in world that was perfectly captured by such a formal system then the perfect model would in fact be a copy of the universe. But what reason would we have for assuming that if the world was mathematically structured, it would be such a simple structure and not one capable of recursion?

The point of a model (perfect or otherwise) is to provide a set of rules that can then be instantiated, and mathematics definitely allows deeper models than elementary geometry. Most rich structures are capable of self-reference, and encoding their own rules: this is a basic observation on which Godel’s incompleteness theorem, or universality of computation rests upon; it can also serve as a launching point for fun programming concepts like quines. These systems are capable of describing themselves completely without any error or trivial mile-for-mile representation. Hence, we can’t use the map-maker argument to defend our rejection of a perfect model of the universe.

Building on the theme of no unnecessary assumptions about the world, @BlackBrane suggested on Reddit a position I had not considered before (and that prompted this blog post) for entertaining the possibility of a mathematical universe:

[Box’s slogan is] an affirmative statement about Nature that might in fact not be true. Who’s to say that at the end of the day, Nature might not correspond exactly to some mathematical structure? I think the claim is sometimes guilty of exactly what it tries to oppose, namely unjustifiable claims to absolute truth.

Is believing that the universe is inherently not mathematical as presumptuous as believing that it is? I agree that we definitely should not rule it out on ethical grounds before we explore it more completely. Believing that the universe is inherently mathematical does give a lot of hope and courage to theoretical physicists. However, if we assume that it is, what does this really mean for Box’s aphorism?

If the universe is a formal system then we (and all our thoughts and hence theories) are just derivations in that system. To avoid the map-making argument, we have to assume that this system is capable of self-reference. Unfortunately, Godel showed that such systems are also incomplete: there are true statements about such systems that are not derivable within the system. As such, no theory we are capable of building as entities inside this system can resolve certain questions that we can ask within the system about the system. If a model can’t answer certain questions, is it wrong? Is it useful? Alternatively, if we are beings that evolved in this system then is there any reason to believe that this would necessitate the evolution of mental facilities that are capable of comprehending the system? There is no reason to believe that evolution encourages perfect understanding of the environment, or even rational learning or decision making about the environment. In other words, even if the universe was inherently mathematical, Box’s quote could still be true of achievable insilications.

Finally, this segues to a third option on insilications that is embedded in algorithmic philosophy and stems from my belief that mathematics is a reflection of our consciousness. We can suppose that the external world is inherently not mathematical and thus not fully knowable or comprehensible, but that all our theories or thoughts about it must be. As such, these theories can always be formalized and one of them might be the best possible description of the things-in-itself that we are capable of thinking. We would not be able to comprehend any deviation from such a model, even though these deviations would exist (in some sense of the word). Would such a model be wrong? Of course, as with the discussion on completeness before, there is no reason to believe that even if such an ideal model existed that we could arrive at it. There is also no reason, to favor this view over a mathematical universe, except in that it is assuming limitations of us and not properties of the thing-in-itself.

Box, G. E. P. (1979). Robustness in the strategy of scientific model building. Robustness in statistics.

From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

### 44 Responses to Are all models wrong?

1. I think you are missing out on some ideas on complexity. This is strongly revealed in your conclusion: “We can suppose that the external world is inherently not mathematical and thus not fully knowable or comprehensible, …” What makes you think that something mathematical is comprehensible? You already invoked one simple form of incomprehension: undecidability in computing. Why stop there? Certainly, things can get so complex that no single human brain can verify correctness. Sticking to the domain of mathematics, we have, for example, the proof of classification of groups, which is estimated to be 20 thousand pages long. Although it has now been written down, and mostly published, its not been verified. Efforts are underway to encode it in a machine-readable format, so that a computer verification can take place, via “proof theory”. Thus, we have something that is eminently mathematical, and thoroughly incomprehensible.

Things get harder from here. How can one verify that the proof assistant itself is free of bugs? We can’t just run it through another, smaller, simpler program, to verify that its bug-free. So the proof assistant itself has to be hand-checked by humans, which is non-trivial (X number of years to get a PhD in proof theory, Y number of years to read and understand the code in the proof assistant, Z number of years to get so very comfortable with it as to proclaim it bug-free. May require more than one human to accomplish this.)

As to a belief that the universe is not “mathematical”: well, what else could it possibly be? Many mathematicians define mathematics as the sum-total of all possibility; to say that something isn’t mathematical is tantamount to saying it isn’t possible. Since there is nothing else that it could be, by law of excluded middle, it must be. (Yes, there is a very fruitful branch of mathematics that explores what happens when the law of the excluded middle is rejected: all of our best proof assistants come from that theory.)

And then there’s physics: shall we assume that it takes an infinite number of decimal places to model the universe? Surely not: we know that the gravitational attraction of that many bits would create a black hole. Ergo, a finite number of decimal places suffice, which suggests that the universe is finite. And if its finite, doesn’t that mean its describable? Feel free to invoke “Kolmogorov complexity” at this point, and attempt to argue that the shortest description of the universe is equal to the total number of bits in the universe. That may be a fruitful avenue: the question then becomes how we, as humans, experience those bits: as identical atoms, as nearly-identical bits of sand. Or perhaps with an AI slant: how does a neural network convince itself that a proof is correct or incorrect? That a proof assistant is correct or incorrect? That a model is correct, incorrect, or useless?

• Thank you for the comment!

You make a good point about practical complexity, but that is not something that I wanted to address in this post (otherwise I would wander into an unreasonably long tangent on computational complexity). I mean to address the question “in principle” and not as far as “comprehensible by a human” but “comprehensible by humanity”. Of course, both concepts are extremely slippery.

As to a belief that the universe is not “mathematical”: well, what else could it possibly be? Many mathematicians define mathematics as the sum-total of all possibility; to say that something isn’t mathematical is tantamount to saying it isn’t possible.

Just because you can’t imagine something, doesn’t mean that this thing can’t be. Even if something is unimaginable by all of humanity, it still doesn’t necessitate that thing being impossible. That is exactly the point I was making in the last paragraph of my post. Like the mathematicians you describe, I personally define mathematics more or less as the sum-total of all comprehensible possibility. However, I don’t have the anthropic arrogance to then go on and project any limitations of human comprehension onto the external world. That being said, there are definitely plenty of mathematicians and physicists that are willing to do that.

And then there’s physics: shall we assume that it takes an infinite number of decimal places to model the universe? Surely not: we know that the gravitational attraction of that many bits would create a black hole

Please let me know if I am misreading your point, but here I think that you make a series of mistakes that I have seen made many times, so I thought I would point them out:

[1] Nobody said these bits have to be localized in one place. The metric tensor in GR, for instance, encodes information about the global structure of the universe, not just local properties. As such, it can be encoded in all of the universe, and there is no reason to postulate that the universe is of finite extent. Thus, even though locally the information density is finite, it is then summed over an infinite universe to encode the metric tensor. If you don’t allow an infinite universe then you’ve established a much more trivial sounding statement: “if the universe is finite then some quantity about it is also finite”.

[2] You also proceed to assume that the parameters describing the universe have to be physically encoded inside the universe they are describing. Although this is required for us to know them, when we are in a discussion of “are they knowable?”, we can’t simply assume that they are, especially not at any one particular moment in time.

[3] Finally, you mistake the map for the territory. If we want to discuss if the universe is describable mathematically or not, we can’t base that discussion on the properties of our current mathematical description of the universe, because then we are begging the question. Of course, this does make the discussion very difficult, but that’s the point! If it was easy then it wouldn’t be fun to think about.

• Hi Artem,

I’ll try to be brief. I think that “practical complexity” also puts bounds on what’s possible “in principle”. This is not an idle remark; you seem to imply that something can hold “in principle” even if it takes an extremely large number of steps to get there “in practice” as if “in principle” is a kind-of extrapolation to infinity from the finite. Let me remark that this kind of extrapolation is very non-trivial; it is only safe to make in high-school pre-calculus and a few other cases, where a “limit” is narrowly defined. In the other cases, practical complexity dominates, and what happens “in principle”, at the limit, is an entirely different “interpretation” (in the sense of ‘model theory’).

Regarding “knowable by humanity”: But humanity is of finite size, too: there are only so many brains on this planet, and each brain is finite in size. The knowable cannot exceed this capacity. We may someday build computer brains that are larger, but they will remain finite in size. There will always be problems that are more complex than what something of finite size can comprehend (or so is my belief: as otherwise one gets a strange situation, where, once you are smarter than a certain level, you can then know everything, ever!? Crazy-talk.)

Re: your point [1], I am confused. The universe is not infinite! It consists of about 10^80 atoms grand total, as measured by astronomers. There is no such thing as “infinity” in physics. Perhaps our knowledge of physical laws is incomplete, but there is no shred of evidence that anything physical could ever be “infinite”.

Regarding the confusion of the map and the territory: Well, yes, this is confusing. But I think you duck another question of principle: what does it man “to know something?” In practice, it means “something that humans can articulate and discuss”. Perhaps there is a better definition, one that could be applied to potential future super-intelligences. But, ultimately, there is a constraint: any super-intelligence will be limited to the 10^80 atoms in the universe available for conversion into a computer (unless Nick Bostrom is right ..) So once again, issues of “practical complexity” dominate.

2. Hmmm, no edit-button to correct my post. Some footnotes, then:

* Proofs are, by definition, decidable. Thus, the proof of the classification of groups is decidable; its simply incomprehensible by a single human, although individual humans do understand parts of it. But this is possible only because our economic system is just large enough to allow a sufficient number of mathematicians to do so. Economics and politics matter.

* Box’s quote is kind-of the mirror image of Kolmogorov complexity, which states that a model is useful only if it is smaller than the thing being modelled, and, what’s more, that there are things that cannot be modeled.

* Cryptography certainly demonstrates the frailty of arguments based on Kolmogorov complexity: encrypted messages are incomprehensible in a certain very strong sense. Based on naive Kolmogorov complexity, an encrypted message is incompressable, and the shortest description of that message is a repetition of the bits in that message. But if you know the secret… then all change.

* “How do we, as humans, experience those bits?” I forgot to ask: “Do we experience those bits holographically, via AdS/CFT?” The point being that the nature of experience is hardly obvious, never mind the philosophical arguments about qualia. We experience the universe by means of the models constructed within our brains, including the rudimentary models in our motor cortex that subconsciously deal with the position of our bodies, and any nearby flying footballs.

* More to the point: “How do we, as humans, experience those bits?” Do we perhaps live in a cryptographic universe, where some knowledge is hidden away in such a way that we need to “break the code” to understand it? (viz, break the code, to create a model of it? In a certain sense, isn’t that what a model is: a decypherment of the messages of the universe?) Perhaps there is a much, much shorter proof of the classification of groups, but it will require more than 10^160 bits of computing effort to obtain it? Which is more bits than there are in the universe?

3. Marc Harper says:

Great post Artem! This is one of my favorite things to think about. IMO, all scientific models are “wrong” for the simple reason that we have no axioms for The Universe. Sure there are many things that science believes to be universally true, like causality, but we don’t really know, and can’t ever know. There are recent indications that entanglement occurs in time as well as in space. People once thought Newton has discovered the universe but we’ve seen so much more since then.

Every model — indeed every idea in every area of human thought — is provisional. Each notion has a limited domain of applicability and is predicated on assumptions. No assumption is universally valid, and even if it were, we can never prove that it is, regardless if the universe is mathematical or not.

Scientific models, however, follow an evolutionary paradigm — models are “less wrong” over time (say on average, built-to-be-wrong models and people’s pet models notwithstanding), in that they fit all the available data better. They make better predictions — they increase our prediction power over time. So they are less wrong in the sense that our predictions are more accurate and precise. Those models get replicated and extended. They mutate into new models that proliferate better. Then we poke at the mathematical holes in the latest theories, looking for experiments to break the models. Sometimes the model is so good that it points to something new, like when Dirac predicted the positron. Sometimes the landscape changes and a new type of model emerges from the void. But a model never becomes “right”, it just aspires to “fit all known data”.

As a mathematician, I don’t consider proofs to be models in this sense. Given the assumptions, certain logical outcomes are valid, full stop (assuming the proofs are correct). Whether the assumptions are really true is another story altogether. To me modeling and proving things about models are complementary activities, and there is an art to both.

I like to think about evolution as an automatic or default process. Some things proliferate better than others, and in doing so, encode information about the environment into their structure, be it a population distribution or an internal structure like DNA. But there’s nothing that says that evolution as a natural process necessarily leads to a maximal conversion of environmental information over time. Real replicators seems to act more like dumb replicators than best-reply-computing smart agents. That’s partly why I find the relationships between information theory and the replicator dynamics so interesting!

It is also interesting to consider the possibility of an ideal model that explains Everything (TM). All things that replicate do so with variation, and if some hypothetical replicator did not naturally diversify, eventually another replicator will out compete it, unless the unvarying one were somehow “perfect” already. This suggests that evolution will never stumble upon the “ideal model”, or the ideal model doesn’t participate in the evolutionary process. Or perhaps we live in the part of the universe that evolutionary things live in, and the ideal model lives in some other slice of the universe, where maybe everything isn’t finite, and Godel doesn’t apply. In the end, I suspect this is a timeless philosophical question — there will always be things we don’t know and always people wondering about if there is some model that explains the unknown or ties together the known.

Thanks for posting!

• Extremely good points. I am a big fan of evolutionary explanations of culture, so I am inherently happy with your evolutionary theory of scientific progress. The frightening thing about evolutionary processes (and other myopic optimizers) is local maxima! But I guess we do see them in the history of thought, I guess Kuhn’s paradigm shifts would correspond to crossing small fitness valleys.

I think the sociology of scientific progress is fun to think about, but not exactly what I wanted to touch on in this post. In your framework, I think what I was asking was: does a global fitness peak exist? If it does exist, is this a consequence of the universe or the capacities of humanity?

I don’t think I understood your last paragraph, though. In particular:

All things that replicate do so with variation, and if some hypothetical replicator did not naturally diversify, eventually another replicator will out compete it, unless the unvarying one were somehow “perfect” already. This suggests that evolution will never stumble upon the “ideal model”, or the ideal model doesn’t participate in the evolutionary process.

I don’t see how the second sentence follow from the first. The ideal model could be present (or could not; I am not advocating that it is, I am just asking that the logic of the argument be clarified) and then we would just get stuck around the fitness peak. Not currently being at a peak is no argument for it not existing, as I tried to show before, just because (even local) peaks exist, doesn’t mean that we can get to them.

i agree that the relationship between information theory, computational complexity, learning, and evolution is very interesting! I will actually be emailing you about this soon, but that’s a topic for another day.

• Marc Harper says:

I certainly agree with your point — not being at the peak doesn’t imply that the peak fails to exist, and you are correct that my second sentence doesn’t follow as an implication of the first. I was trying to say that an “ideal model” would have no need for a variation and selection process (since it’s by definition the pinnacle of models), and it ideally would not vary at all, since that would only cause the model to drift from the peak. So if the ideal model was reached by an evolutionary process, it/we would have to taper off its variation somehow, assuming we had a way of knowing it had reached the peak, which seems unlikley; if this assessment was made prematurely, as has happened many times in the history of human knowledge, some other model with variation would eventually overtake). Or we would have to allow a “quasispecies” of near-ideal models to be synonymous with the ideal model, but the we’d not really be at the peak, and there would be some chance of drifting away.

Perhaps I’ve stretched the analogy a bit too far and I’m not working with precise definitions, so I’ll stop here…

• Sergio Graziosi says:

Quote: “Every model — indeed every idea in every area of human thought — is provisional. Each notion has a limited domain of applicability and is predicated on assumptions. No assumption is universally valid, and even if it were, we can never prove that it is, regardless if the universe is mathematical or not.
[…]
They make better predictions — they increase our prediction power over time.”
Thanks Marc, this is essentially what I’m exploring over there: http://wp.me/p3NcXb-1d (warning: long read into multiple posts).
Of course all models are wrong: can we measure anything without approximation? No. Can we include all variables into a model without approximating their value? No. Can we proof that we know and can measure all variables? No.
Models are approximations, hence are inherently guaranteed to be somewhat wrong.
What I find interesting is observing that the usefulness of a model is usually negatively correlated with its precision. Silly example: more or less all of us rely on a simple physics model that predicts that if we let go an object in mid air, the object will fall to the ground. Useful, but fantastically inaccurate, if one looks at the whole Universe, the places where this “prediction” is reliable are tiny minority. But that doesn’t matter, because the simple model is reliable where we happen to spend our lives.
So yeah: all models are wrong, but we can still use them and find better ones. A model is a useful tool, nothing more.
On other discussion here: models don’t need to represent reality in a “true” way. For example, Linas says: “The universe is not infinite! It consists of about 10^80 atoms grand total, as measured by astronomers.”
Uh, yeah, but how can we tell that there is no cosmic field that isolates us from another even larger pool of atoms and from all their measurable effects? We can’t. But we don’t need to care, if 10^80 atoms are all we have access to, we’ll deal with them alone, but saying that therefore the Universe is finite is taking it a bit too far. You are confusing the useful model with reality. Something lots of scientists do, and shouldn’t, IMHO.
Thanks to Artem for the thought provoking post!

• Thank you for the comment!

I feel like the main point of my post is being missed whenever I see these definitive answers:

Of course all models are wrong: can we measure anything without approximation? No. Can we include all variables into a model without approximating their value? No. Can we proof that we know and can measure all variables? No.
Models are approximations, hence are inherently guaranteed to be somewhat wrong.

What I find interesting is observing that the usefulness of a model is usually negatively correlated with its precision.

This is a very interesting point when it comes to heuristic models. Some of the trade off is trivial: (1) a more precise model is usually more complex and hence harder to use, and (2) the ‘increases’ in precision usually come in domains that the preceding model didn’t care much about, and the reason the preceding model didn’t care much about it is because that domain matters in fewer settings (for instance, this is why engineers use classical mechanics and not quantum mechanics when building bridges), but some trade offs are related to things like generalization and over-fitting. Often placing external restrictions on your models makes them better, even if those restrictions don’t have a particularly sound basis.

Linas says: “The universe is not infinite! It consists of about 10^80 atoms grand total, as measured by astronomers.”

I think that he was mistaken “observable universe” for “universe” here. I didn’t continue that thread because I think we are talking past each other there, and because I find the “but its finite” a particularly boring direction for arguments. In particular, the other part of Linas argument relied on Kolmogorov complexity (which he also seems to misinterpret in his discussion of cryptography, assuming that he is referring to things beyond one-time pad) which is defined in terms of Turing Machines. If you go the ultra-finitary root then you have to throw away TMs (since they only make sense in idealizations where you assume arbitrarily large integers are allowed) and replace them by DFAs or BDDs and build an information theory on top of those.

• Sergio Graziosi says:

(trying the ‘blockquote’ tag as simple text, let’s hope it’ll work). Quote:

The whole point of my post (apart from getting me out of a no posting slump) was to point out that one shouldn’t be so quick to make universal assertions. Philosophy is only fun if we try to give respect to all reasonable arguments and explore how far they can be taken. Asserting definitely “No” to the answers you ask just seems close minded.

Oh, apologies for not “getting it” and thanks for the clarification. I see your point of the fun of Philosophy (and the parallel with mathematics is neat). Within the model I’m trying to build (and testing it along the way, thanks for the chance!) I can see where the close mindedness impression comes from, but I can fit it in my view without effort.
Because models make sense only in the context on how we may use them (your mention of engineers and classical mechanics being an obvious case), you are absolutely right that definitive answers spoil the fun, but in my case fun is merely a necessary side-effect (necessary because I wouldn’t be writing this if it wasn’t fun), not the main reason why I’m trying to build my own model.
Also, my kicks come from the realisation that even after asserting a definitive “no”, I can still find out that it’s possible to know something (it’s post three in the series, I believe) and to what ultimately led to sketch my own scientific epistemology, and my own approach of the demarcation problem (that you’ve read & commented already, thanks!). All of the above is philosophically interesting to me and in general accordance to what is being discussed here (absolute claims can only be made about conceptual entities).
We agree on model trade-offs between precision/usefulness and on your assessment of the other comments, so much so that now I fear I’ll become boring! ;-)

• Heh, The nice thing about a physics education is that you learn how to detect sloppy thinking and cranky ideas. One is instilled with a certain hard-headed pragmatism that is often lacking in other disciplines. There’s a reason for this: when dealing with models in physics, either they work, or they don’t, in a pretty clear-cut fashion; nature is a harsh mistress. It is very different in other disciplines, such as biology, or anything dealing with complex systems, where there is a vast abundance of mostly-good-enough models to choose from.

So, for example: trying to insist that the universe is “infinite” is a typical crank-theorist hobby. Basically, its total bullshit. We can see the universe, and we can measure what we see. Insisting that there might be “something more” than what can be seen is just nuttiness: its a belief in ghosts, a belief in the existence of something that can’t be seen, can’t be proven. Now, if you are a credentialed physicist, it is OK to develop models based on infinite universes, just to see how they might work out: at the heart of theoretical exploration is the need to explore sometimes fantastic-sounding, cranky directions, just to see if they work. But unless you can communicate your ideas in a language that other physicists will accept, you will be labelled a crank. The conception of an infinite universe is an example of crankiness.

There are only two exceptions to the above: one for philosophers, and one for mathematicians. Philosophers are allowed to grapple with totally crazy ideas, because it is their job to convert vague, unclear, confusing issues into something sufficiently clear that ordinary science can take over. The exemption for mathematicians is made because they have the tools for talking about infinities, via axioms such as CH, Martin’s axiom, large cardinals, and the mechanics of “model theory”, which show how to work with “non-standard interpretations”. To insist on infinity outside of this context is .. crazy talk.

Well, but that’s OK, since I took Artem’s original post to be philosophical hand-waving and vague ideas about poorly-defined topics, and not some kind of rigorous development. But, as physicists know, you should only incorporate crazy ideas into your models if they actually make the models better, stronger and more accurate. If they don’t, then crazy ideas are will only clutter up your mental space, and prevent you from perceiving reality as it is.

4. Boris Borcic says:

Strikes me most here the name of Box himself – that name works as the IMO most concise allusion thinkable to the idea of reforming the understanding classical-physics-turned-elementary persists to inspire successive generations of newcomers, that physics models should conceptually BOX the isolated systems they model – hold them like prisoners…

…reforming that understanding by something like an inversion that transforms almost-perfectly-predictive-models-of-almost-perfectly-isolated-systems into vanishingly-constraining-normative-models-of-vanishingly-connected-systems…

…and transforms the induced thirst for unification into a reappraisal of perturbative modelling.

(Hum, at the first failing attempt to post I discover that a profuse discussion took place in comments since I’d opened the page in a tab – posting anyway, please forgive serendipitous interferences as may occur)

5. James Rose says:

I apologize in advance for taking a simplistic turn – (though my interest is in Unified Theory of Behaviors (versus ‘unified field theory’) not simplistic at all) – – But I pose a question:

Here are two equations:

E=mc^2
F=ma

Are they each ‘models’? Are they ‘theories’? They are both testable, with one easier and more accessible than the other, but within test-condition-variables and statistical acceptability,
do we really want to say, as Box has, that “all models are wrong”? He is suggesting the extrapolation : “all equations are wrong”. Is that that where we really want to go?

• linasv says:

A single equation is like a factoid: “48% of everyone are conservative” (0.48*E=C): rather meaningless without the extensive accompanying corpus that explains what “everyone” and “conservative” mean. In essence, then, a model can only be a large, mostly-consistent network of facts that conveys some sort of meaning, and approximates some sort of truth. Since anything we could ever say is an approximation of truth, then its self-evident that all models are wrong. However, this begs the question: what is meaning? What do the words “fact, network, truth, approximate, consistent” really mean?

• James Rose says:

Ahh Linas, the mire and muck of ‘semantics’. A truly scary purgatory to explore. But, i suppose,explore it we must. My first observation to your thoughts is — do you really want to trivialize e=mc^2 as a ‘factoid’ ? My next is to share with you my touchstone frame of reference, the linguist Benjamin Whorf ca 1930s “Language, Thought and Reality”:
“We are thus introduced to a new principle of relativity, which holds that all observers are not led by the same physical evidence to the same picture of the universe, unless their linguistic backgrounds are similar, or can in some way be calibrated.”. Information, data, meaning . . indeed require ‘relational calibration/coordination . . sufficient mapping’. And since the universe seems to cross function interrelationally through many levels of organization quite smoothly and nicely, there is at least the hint of a fundamental pervasive and persistent cross calibration basis .. everywhere, all the time. So, the ‘truth’ is out there, waiting to be meaningfully identified (interpreted) . . calibrated, so to speak . . with whatever our capacity to appreciate and understand about it. May I ask you also — you wrote: “Since anything we could ever say is an approximation of truth, then its self-evident that all models are wrong”.
Wow! any error = total error ?!

• Well, E=mc^2 is a factoid in the sense that the ‘actual’ eqn is E^2 = m^2 c^4 + p^2 c^2 and the last is obtained only for p=0. The ‘actual’ eqn itself arises an invariant of the Lorentz transformations, and there is a general theory for such invarients in a general setting of Lie groups and (pseudo-)Riemanain geometry. I forget what kind of invarient it is, maybe a Casimir invariant or something like that. Oh, and the invarience doesn’t hold for off-shell particles in QFT; instead, it appears as the denominator of the propagator (more colloquially, the Green’s function, or Fredholm alternative, depending on your point-of-view). (see for example, wikipedia article for the Klein-Gordon eqn, where you’ll see Einstein’s famous equation with Einstein nowhere in sight.) Well .. and even then, that’s just for perturbation theory. The non-perturbative aspects remain a (hot?) research area. Anyway, that’s the tip of the iceberg of what I remember from grad school.

I read Benjamin Whorf as an undergrad, pretty sure I read all of ‘Language, Thought, Reality’, and remember none of it, at all. I desperately wanted to major in AI because physics was so booooring, but AI was not yet, at the time, something you were allowed to study (at the undergrad level).

So I took the long road, and study semantics these days. And linguistics. And machine learning. And category theory. And type theory and model theory. And get paid to convert some of those high-falutin ideas into crappy computer software. Alas and alack, it is indeed purgatory for some past sins committed. But yes, I suppose I agree with Whorf and Mulder, ‘the truth is out there’.

• James Rose says:

Thank you for the reference frames, Linas. The inter-relationals hint strongly that there is indeed a pervasive way to umbrella all of them together in a meaningful way. More than a unified theory of fields (topologies) . . a unified theory of behaviors (as I identified here previously). Unfortunately, current math is missing critical topos transforms, and, there are 400 years of concepts biases that give priorities to particles and forces without “explaining” them – simply describing them. What if I were to suggest to you that the core similarities (which are seen but incorrectly modeled), are general density/intensity relations -first-. That is, behaviors/least actions are induced by gradient differentials first, and that the ‘fundamental forces’ and thermodynamic entropies are example forms of a priori topological relations that precede them. Currently, conventional theories identify entropy only as thermodynamic, being the impelled product of the ‘forces’. The truth is already here where we can see it: primal geometry – interrelational dimensions – understood for density/general entroepic relations – precedes the fundamental forces and thermodynamic entropies. The 4 & Therm are sample generated forms of General Dimensional Entroepies; The 3 Newtonian Laws are secondary and educable from general entroepic principles as well. BUT, and I write this with a very large grin and smile, This exposition doesn’t work without a key new mathematical argument of information coding~transformation. (hinted at in the fields limits of recent string theory where critical formulae factors reduce to division-by-zero states; a wall of meaninglessness reached).

A total new interpretation of dimensional coding is needed – akin to “where does the wave information go to when wave functions ‘collapse’ “? Critically, there is re-constructable re-constitutable information present, even if a state goes to ‘zero’. Otherwise, how can we rationally talk about a phenomenon called ‘symmetry breaking’, where zero data suddenly equates with vast non-zero measures and relations?!! Doesn’t anyone realize we can’t just do hocus pocus math without explaining and justifying it !?!?.

Please reach me at integrity@prodigy.net if you care to discuss things further. :-)

The fascinating thing about language and semantics is that even the most ethereal intangible memes, words, notions, concepts – can all be traced back to tangible phenomena through the etymological pre-tree. Everything. Whorf rules! He specified the core priori relationship of dimensional topos reality … essential connectivity, translational mappings.

6. Zach M. says:

“However, the question remains: is it conceivable that there is a perfect infallible model of the universe? Is the world fundamentally mathematical and comprehensible? As long as I can remember, I believed that the world is not inherently mathematical, that mathematics is a reflection of our conscious and there is no reason to expect that it can perfectly capture external reality in any sense of the word. In fact, I have always asserted that assuming the opposite is anthropically arrogant.”

This is the passage I was referring to in your ‘Kooky Quantum Mind’ post. Kant made this point, and basically said that Euclidean geometry is a facet is our perceptual apparatus and not necessarily a facet of the noumenal reality.

7. Ramsay says:

I will reply here to your last comment on the G+ post where we started a conversation:
Specifically, I said

> I feel like the universe has some kind of unbounded mathematical ‘depth’ that will never be fully captured by any theory we construct.

And you replied:
“I don’t really understand what you mean by this (or the paragraph it starts). Could you expand on this more? Or point me to somewhere where it is explained more? You write that this is similar to my stance, but I don’t really understand how. For me, I think that theories don’t ‘capture’ but create in a lot of cases.”

I am glad that the verb I used was ‘feel’, rather than ‘think’ (or worse ‘believe’, but I don’t think there is much danger of me accidentally using that). For whatever it may be worth, I will try to expand on that sentiment a bit, but there is no well-formed philosophical stance here, and if the ideas have provenance in somebody’s writing, I have forgotten the source.

I read the post you mentioned https://egtheory.wordpress.com/2014/09/11/transcendental-idealism-and-posts-variant-of-the-church-turing-thesis/
and although a lot went over my head, I liked the ideas of the sensible world and the intelligible world. The sentence that you have asked me to expand on involves the concepts of ‘the universe’ and ‘mathematics’, and unfortunately I can’t think of a satisfactory definition of either. Not a good start. I think definitions are really important, but sometimes they are hard to find, and that might be the whole point. Maybe the universe is the sensible world and mathematics is the intelligible world? (I don’t know the definitions of those either.) Whatever they are, I think they are different. At least they are accessible to us through different windows.

In mathematics there is always a push for deep ‘grand unifying’ theories that relate seemingly distant disciplines into some common abstraction. But nobody expects there to be a Theory of Everything, from which all future mathematics can be derived. I guess there maybe was a time, before Godel, but it isn’t in the culture now. In my undefined language, the world of mathematics has unbounded mathematical depth :-)

However in physics the dream of a Theory of Everything still seems to live. I hope we see a viable Grand Unifying Theory in my lifetime. I am optimistic (for no good reason) that one will be found, but I don’t expect it to live up to the dream of being a ToE. I find it surprising that so many great physicists do still seem to anticipate a ToE. I imagine that the new theory will significantly alter our perspective and enable us to obtain GR and QM as good approximations in the appropriate limits. I guess the hope is that since almost everything can be explained by either quantum mechanics or general relativity, then a theory that encompasses them both should explain everything. This just seems really naive to me. (can I say that without sounding arrogant?) I expect that any new theory will raise fundamental questions that we have not yet even begun to conceive. But they will be questions that we can hope to answer, and so the quest for a grander theory will resume.

I do not really have arguments to support this expectation other than going back to the assertion that there is a distinction between the universe and a mathematical model of it. I should probably read David Deutsch; it seems like his viewpoint would resonate with me. If a ‘mathematical structure’ is something that admits a finite description [that is maybe too limiting: something that is ‘comprehensible’ anyway], then my feeling is that the universe does not correspond exactly to a mathematical structure (although I agree with BlackBrane’s point that this possibility can’t be ruled out). However, by any reasonable definition ‘mathematics’ does not correspond to a mathematical structure either. I do not want to say that mathematics is inherently not mathematical :-) I do not want to say that the universe is inherently not mathematical either.

Your definition of mathematics as the “sum-total of all comprehensible possibility” could work for me if we took a sufficiently broad interpretation of ‘comprehensible’ and agreed that anything capable of comprehension is not capable of comprehending all of mathematics. I want the definition of mathematics to be independent of the human experience of it; I have a Platonic viewpoint. Mathematics is unimaginably huge, and other beings may have a dramatically different experience of it, but if we are able to communicate with them it will only be because we have explored common regions of mathematics. My imagery is more geometrical than algorithmic. I recognise that we ‘create’ things in mathematics, but the biggest things, the deepest things, we find.

So if we say ‘mathematical’ means ‘of or pertaining to mathematics’, then mathematics is mathematical even though it doesn’t correspond to a mathematical structure. I feel that the universe is also mathematical.

I see the universe as an objective reality, independent of our experience of it. Our perception of it has undoubtedly guided our exploration of mathematics. From our perspective, the universe appears to be much more constrained and limited than mathematics, but there is no reason that I know of to believe that it is so simple as to yield to a Theory of Everything. I would want to define the universe in the broadest possible sense, independent of our experience of it. It is the sum total of all objective reality, not just some piece of a ‘multiverse’ (so ‘multiverse’ is just a name for the universe in some cosmological models). I am not just trying to hedge my bets there: obviously we can never hope to have a viable theory of a reality that is completely inaccessible to us in principle. I think that any fundamental theory will reveal cracks in our knowledge which we can explore from our corner of the universe.

I wrote that I thought that my philosophical viewpoint was similar to yours. I should not have said that: I do not have a clear understanding of my own viewpoint, let alone yours. But at least I think we are in agreement with the “all models are wrong” sentiment. Your remark that “theories don’t ‘capture’ but create in a lot of cases”, indicates that there must be significant differences in our outlook, because I have thought about it a bit, and I still prefer the verb ‘capture’. I feel like our insilications give us greater insight into the universe. They may open up our experience to new vistas that we didn’t even imagine before. But these aspects of nature are ‘there’ independent of our discovery of them.

I have enjoyed reading your thoughts on these things. I don’t usually engage in philosophical discussion; I get overwhelmed with a feeling of hopelessness. But this post of yours really resonated with me and maybe brought out some latent philosophical tendencies.

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