Shut up and calculate: an extreme mathematical universe
arxiv.org
arxiv.org
But it's also a contradiction since science scientific methodology demands we check our results empirically: the fact that you have subjectivity (i.e. that qualia exist) and what it's like, is the most direct empirical fact we have access to and which needs reconciling in any discussion of the universe's constitution.
I think people sort of push this point aside either by saying that qualia/subjectivity could be generated by math (or a computational process), and that's what really matters. But we have no examples of abstract math generating anything; physical processes generate things and we use math to influence their behavior—but that's it! And furthermore, even if math generated subjectivity—it's still there in all of its un-mathematical obviousness at the end of the day.
I've yet to see an argument in favor of math having higher ontological status than this which wasn't every bit as unrestrainedly imaginative and unfounded as Plato's 'ideal forms'. I would expect a thoroughgoing adherent to scientific methodology to instead say something on the lines of, "this is outside the domain of science to judge one way or the other; the most we can say is the idea that reality is made of math is one untestable hypothesis among many whose probabilities of correctness each approach zero by dint of their many peers."
If simulated universes can be "real," then we're assuming that there's a mapping between patterns of electrons in a computer (or provable theorems, if you're brave enough), and the real universe they represent. (Say, interpreting two numbers that may be stored very far apart on the chip as the positions of two particles that are, in the universe, very close together.)
The problem that this introduces is that there are no well-motivated ways that we could decide which mappings are "real," and could be "lived in." For example, we might claim that the state of the simulation must be map-able in polynomial time to some charts and pictures that a human could interpret - but an inhabitant wouldn't care.
So, since all mappings are equally motivated as far as an inhabitant is concerned, allow me to choose one that maps this grain of sand to a universe the same as ours.
Well, there you go. Our world in a grain of sand.
Even if there might not be a single most natural mapping, I think it makes sense to say that some mappings are more natural than others. I think there is a natural sense which is independent from human perception and intent, in which a computer running, say, a graph coloring algorithm, is doing that more than a waterfall is.
Even if there is a bijection between a collection of initial states of the waterfall with the input of program, that fits together with one between the outputs and the results of the waterfall.
I agree with this part.
I wasn't thinking specifically about P vs NP though. (if I was, I think that would be a kind of weak argument because we don't have a proof that P isn't equal to NP ? At least not yet.)
I was thinking more in terms of like, Solomonoff probability and Kolmorogov and things like that.
And even if you suppose that the universe in question has a Turing oracle, I think you would just have to go up enough levels in that hierarchy in order solve that.
Even without that though, looking at the space of permutations on N, I think it is likely that there is a natural way to talk about what things are true of "most" things in that space, and things like that, which could, I think, support at least a weak sense of degrees of naturality between mappings, even without appealing to computational complexity.
Edit: I figured I should respond to this part also:
> In fact, if the problem is right, you might even end up in a situation where they can't hook their NP-hard problem solvers together in a way that solves problems that we would consider polynomial.
I think that whatever process is needed to get the output in a form that can be used as input for other things in general should be considered to be part of the computational process. Maybe that should even be implied by the meaning of "compute".
Like, say that some computation takes a number as input in a binary representation, and outputs a number, but outputs it in ternary. If you are talking about how fast the process is on numbers, it should include the time needed to convert back to the binary representation. (Unless, of course, the computation that you are considering the process to be doing includes the "input is in binary, output is in ternary" part. If it is being considered "on numbers" though, I think it should include the conversion time to and from whatever representation of numbers is being considered to be canonical.)
The fact that you have to consider the complexity of the encoding scheme and not just the input and the output bit patterns is well understood in information theory. If I have an encoding scheme that has included in it a complete simulation of the universe, I can encode any video into 0 bits since the decoder will always know what I am about to request of it. The more the mapping is customized to the information you're searching for, the less complexity is needed in the input bit patterns to produce the desired output. If you go searching PI for a message, it'll probably take you longer to find it if you use the ASCII values of characters than if you use base64. You'll find it even quicker if you specify that 3='h' 1st 1='e' 4='l' 2nd 1='l' 5='o'.
This all ultimately comes back to Ockham and parameters on a model. Models with more parameters can fit more things spuriously, so you want the model with the fewest parameters that can do the job. Minimum Description Length (and MML) is a great concept https://en.wikipedia.org/wiki/Minimum_description_length
If your mapping is very complex compared to the ostensible simulator, then the mapping is actually doing most of the simulation. So the simulation is not mostly running when the ostensible simulator runs, it is running when you perform the mapping.
If you are inside the universe being simulated and think you can do that mapping, it seems unlikely that there exist enough time and space for it.
But, what if there's a law of physics inside of the simulated universe that solves exponential problems in polynomial time? You could pack a more limited universe inside of that one, if you didn't want your inhabitants to have access to it. I think that computational complexity absolutely does determine how useful a simulation is to us, but it's not meta-universal enough to do this kind of philosophy with.
If I'm interpreting it correctly it does seem to put some constraints on the nature of the simulating universe, but does not necessarily disprove the existence of it?
We can divide up the task of running a physical simulation into the part where the state is computed, and the part where the results are made visible to us in a way we understand.
We can also divide computation up into the same two pieces: the actual solving of the problem, and then the task of reading symbols off the Turing tape. Usually, we set up our definitions so that a program isn't thought to "solve" a problem unless the reading-the-answer part is trivial.
Now, for a person living inside a simulation, the reading-the-answer part would become insignificant. Why do they care if we can see in to their universe?
So, if all we want is to make a world for simulated people to live in, why not stuff all of the computational complexity into the reading-the-answer part, and then not do it? They would be perfectly happy with a jumbled-up, uninterpretable state in our world, because they would be living on the inside where it made sense.
So, if that is convincing, then allow me to pitch my universe simulating computer:
Launch two photons into space. The distance between them is the state of the simulation. If you want to look in to the state, you have to solve an incredibly hard exponential problem to map the distance to a certain corresponding wavefunction that you could make sense out of.
However, the inhabitants wouldn't care about whether or not you interpreted their state. So, as far as the simulated are concerned, two photons flying apart do make a simulation!
In my opinion the easiest way to make this reductio-ad-absurdum go away is to reject the idea that the simulated is real.
However, it's not really that surprising: Science over the centuries has steadily (and for it's time, justifiably) eroded the role of subjectivity in order to uncompromisingly get at what is consistently true (regardless of what we'd like to be true) and this process required being very suspicious of any subjective accounts that can't be measured.
That being said, perhaps the time has come to dig a bit deeper and in the same persistent and uncompromising way ask what does objectivity or subjectivity mean? What do they arise from and to whom? I don't know if any new technologies can be derived from having clearer insights to these questions, but I do feel that it would be time better spent than any untestable mind made artifact about what reality ultimately is.
However, my suspicion is that the experience itself will remain not amenable to conceptual analysis—that it will remain fundamentally incapable of systematizing. For instance what would the mathematical description of the experience of a fever chill look like? (Hint: the definition there requires that you don't take a reductive approach to this: it's not asking about a mechanism which produces or is responsible for the experience, it refers to the experience itself.)
That said, I think it would be very valuable to understand better what the limits of our conceptual faculty are: as impressive as cognition is relative to other things we and other animals are able to do, we're being unduly anthropocentric in assuming it to be boundless in capability. More recently I've come to consider concepts as almost like another sense: they are symbolic patterns consistently formed when we are exposed to certain stimuli (like our experience of smells consistently reappearing when exposed to similar molecules). Our conceptual faculty augments the pure pattern-correspondence of our more primitive senses in that the patterns can be associated with other internal patterns, and in that we can generate new patterns purely from existing patterns (using logical and analogical processes), which at some future time may be usefully associated with some never-encountered external stimulus.
Finding the limits to our conceptual faculty is almost synonymous with find its 'structure', which I think would have clear benefits.
I'm in total agreement with this as well. When I suggested looking into what is it that objectivity and subjectivity arise from, I wasn't referring to the "physical" explanation (which I assume will eventually be resolved, though not in the near future) but to the core of the hard problem of consciousness. My take on this is that when looked at closely, it will not be found within the domain of anything that can be conceptualised simply because it precedes concepts. Any concept requires consciousness to exist but not vice versa. Of course the usual argument at this point is something in the line of "so the sun didn't exist before any thinking animals existed?" To which my answer is that the sun as a concept didn't exist. Neither did hydrogen, electrons or wave functions. Only pure pre-concept reality existed, neither objective nor subjective, "waiting" for a mind to evolve and eventually form subjects. objects and concepts that with increasing accuracy describe how this reality functions, and then proceed to confuse this description of reality with reality...
You have sense organs and sensory cortex that creates representations. Those representations are evaluated subjectively for their utility, thus, generating emotion. This is qualia. What else is there? The fact that it feels like something to see red or blue? That's just sensation + emotional response.
I know it feels like I just gave you the soft problem of consciousness instead of the hard problem, but I don't think the hard problem is anything beyond a loop of perceive - evaluate - act - learn in the world.
If you want the ontological cause of consciousness, then it must be the environment because the environment teaches the agent perception and values. Values and emotions appear from the game the agent is playing. So the substrate of consciousness is not the brain, but the "game".
I think you just pointed out the very paradox which illustrates my point: In which category do you place something that cannot belong to any category?
At this point you might want to say that I'm just making up an artificial nonexistent construct that has no bearing on reality and my task would be to show that not only does it exist, but it is the very basis necessary for all categories to exist. This thread would probably not suffice to achieve that :-) And by the way, I don't see it as supernatural - if anything, the process of concepts and objects created by an agent to approximate reality and then taking these approximations to be reality is what I find to be supernatural. It amazes me how evolution managed to pull this off...
>The fact that it feels like something to see red or blue? That's just sensation + emotional response.
I'm suggesting that "just sensation" is a placeholder for "I have no idea what that is"... There's a big difference between the explanation of a process and the experiencing of a process. We are so quick to categorise experiences into concepts and relationships between them that we overlook the fact that the sensory experience exists before any conceptualisation and categorisation, and in a way is much more real than any story we mould it into some milliseconds later (i.e "this is red", "this is cold" etc.)
>So the substrate of consciousness is not the brain, but the "game".
By now you probably realise (though I assume still disagree with me) that I see it the other way around: i.e that no concept can exist without knowing it. This is not the same as saying nothing exists without our knowledge, but it does mean that what exists before our knowledge of it remains by definition undefinable and inscrutable.
How could something created by a body not belong to any category? It belongs to the category of self-regulated systems adapting to their environment.
Creating concepts has been demonstrated since 2012 (Google 'cat detector' https://www.wired.com/2012/06/google-x-neural-network/). In natural language processing, Andrej Karpathy demonstrated in his char-RNN blogpost the emergence of neurons specialized in various concepts (http://karpathy.github.io/2015/05/21/rnn-effectiveness/). It happens naturally in deep networks, as lower layer features are combined into higher level features.
> I'm suggesting that "just sensation" is a placeholder for "I have no idea what that is"...
Ah sorry, I didn't mean "sensation" as in "placeholder for no idea". I meant it as a neural net, the kind we have in our vision system, or the kind used in Deep Learning. They are made of millions of neurons and trained on millions of images to distinguish and localize objects and object relations in images. We have had thousands of papers optimizing on this kind of architecture - Convolutional Neural Network (CNN) - arguably the most successful part of deep learning. Not only that "sensation" was a neural net, but "emotional response" is also a neural net that computes scores of utility of states and actions.
So I was referring to concrete concepts. That's what I like about RL compared to philosophy - it replaces vague concepts with concrete models.
We obviously mean different things by "sensation": I've implemented neural nets to solve problems using ML, I understand the theory and certainly impressed with the results . I assume that by "sensations" you are referring to the process of inputs entering the input layer of the neural net. This is exactly part of a model for solving the soft problem of consciousness. What I'm getting at is that you know what a sensation feels like before you give it any label - it has a certain "is"ness to it that no equation or concept can convey. I'm saying this becomes possible when a system becomes complex enough to be self-conscious and it is usually attributed as some sort of emergent property that is created by the system, whereas I claim it is much deeper than that, and in fact it is the other way around.
Obviously this correspondence format is not really suited to dive deeper, but I'm always happy to discuss this to find if and where I'm mistaken.
Do they? Everyone is entitled to their unquestioned assumptions as long as they state them outright and describe their parameters. I have yet to see dualists (which ultimately is at the base of this whole line of argumentation) admit that they posit subjective experience _without evidence_, since they equate asking questions and "seeing" things as subjective experience. It's worthwhile debating and deconstructing just exactly what that means before accepting it and its hoaky mystical follow-ons. The fact remains that no entity can successfully prove to another entity that it has inner experience, so how would proving you have subjectivity to _your own mind_ work? You can't, and probably, you don't bother, because it is somehow "self evident" that you have this property. The whole qualia and consciousness debate is riddled with circular reasoning about something that dualists just "feel." Well sometimes I feel like I might have the same thoughts as a jelly donut, go ahead and prove me wrong with math!
> I think people sort of push this point aside either by saying that qualia/subjectivity could be generated by math (or a computational process), and that's what really matters. But we have no examples of abstract math generating anything;
This is a bizarre and unsupportable statement. Clearly physical processes can carry out computation, which is a special case of mathematics (or vice versa, depending on what you got your doctorate in), and computation can produce statements in whatever language you desire, even English. Yes, chips and software stacks produce outputs, even in English. How is that not "generating" something? Ah, but you ask what is inside? Tons! There are gazillions of bits inside computers, and similarly gazillions of flashes in your neural circuitry. That's a whole lot of "inside". How do you know that it _doesn't_ feel like something to be all those twinkling bits? If you still accept without question that you do in fact have subjective experience, doesn't it stand to reason that if the neural network which absolutely definitely does correlate with your consciousness did feel like _anything_, then by definition it would have to feel like this?
If you are trying to 'prove' subjectivity, in the sense of providing an argument for it, then you are missing the point: you already experience it directly by definition, so there is nothing to prove.
> This is a bizarre and unsupportable statement. Clearly physical processes can carry out computation, which is a special case of mathematics ...
I am arguing against the idea that the computation (i.e. the math part) carried out the physical process—not the other way around.
Ah, I see. But that follows directly from the computable universe hypothesis.
I can also perceive that glasses of water magically break and reconstitute pencils, in what we now know to be an optical illusion. "Self-evident" observations stand only so long as they are consistent with all other observations. The premise that subjectivity exists is not consistent with everything else we know without introducing additional structure to the universe that has no purpose but to stroke our egos. Therefore, it should be discarded.
We must be using different definitions of subjectivity. I'm just referring to whatever one experiences at some point in time. Do you experience anything? Then you have subjectivity.
We are, because an ontology that takes subjectivity as primitive/irreducible is different than an ontology that reduces apparent subjectivity via third-party objective facts. You subscribe to the former, I subscribe to the latter. It is precisely this distinction that differentiates materialists and dualists in the philosophy of mind.
This question doesn't yield any insight, because it still leaves us with the question of the nature of subjectivity, a term you use with impunity but without definition!
You seem to be implying that subjectivity and objects must form some sort of recursive relation, but this then already assumes subjectivity is irreducible and the question at hand is precisely whether such an elaborate construct is necessary. There's plenty of evidence hinting that we're just special pleading about consciousness's special properties.
>it still leaves us with the question of the nature of subjectivity, a term you use with impunity but without definition!
I didn't try to promote or define subjectivity, I challenged you to define an object without making any references to something subjective (as this is the ontology you said you subscribe to). What I was getting at is that you'll be hard pressed to present any object that is not a concept created by a subject. Mind you, I'm not saying objects and subjects are real, but that they are co-created by the mind and as such are equally unreal as far as representing what actually is.
I can easily posit an ontology that references no subjects. Just because I formulated said ontology from sense perception doesn't entail that it references subjective concepts.
> Mind you, I'm not saying objects and subjects are real, but that they are co-created by the mind and as such are equally unreal as far as representing what actually is.
Except as per Moore, this sort of skepticism necessarily presupposes the very knowledge that it attempts to undermine, and therefore it contradicts itself and is literally false.
My point is that calling it sense perception will not make the subjective pre-conceptual experience of it go away. I'm not advocating a ghost in the machine here, but trying to show that neither the ghost nor the machine can survive as primitives when put under deep scrutiny, but I first need to demonstrate how neither can independently exist as a valid concept.
>this sort of skepticism necessarily presupposes the very knowledge that it attempts to undermine, and therefore it contradicts itself and is literally false.
Not sure I follow how this relates to the quoted sentence, but the problem with any explanation (physical or philosophical) is that it is always limited to discussing concepts and relationships between them and can never break the boundaries of what can be grasped by thought (this explanation not excluded). My attempt is to show that thought (and thus any object derived from it) is not the pristine reality, but something that comes after reality is processed. At most, thought can point in the right direction (e.g, understand it's own limitations).
You're evading the issue. You initially claimed that there is nothing to prove about subjectivity. This is patently false.
The subjectivity you apparently experience by observation, if taken at face value as you initially suggested, ie. "there is nothing to prove", is logically irreducible to third-party objective facts.
And yet, third-party objective facts are the best explanation we have for everything else.
Therefore, you do have something to prove regarding subjective experience, contra your original claim.
Here's another problem. You know about the physical/mathematical properties of the world by abstracting away from our experience of a world with color, sound, etc in it. Remove the colors and how would you know about shape, extension, mass?
Predictive coding in the interoceptive and control networks of the brain, actually. Sure, it doesn't explain everything, but it pretty well dispels the intuition that makes p-zombies conceivable and explains the vast majority of the empirical and phenomenological facts to be explained.
Could you point to something related to predictive coding and control networks explaining anything phenomenological (or give an argument)? From what I can tell the two are completely orthogonal.
Edit: I should add that my suspicion is you are mistaking a system which is isomorphic to relationships between phenomenological entities with something which could account for the presence of the entities themselves.
I would recommend checking out the comment here from user 'theoh' on the "relations are what matters" perspective.
Because it doesn't exist in a vacuum, it exists inside an agent, that is inside the world. And the agent has to achieve a goal, to maximize utility, by learning to act intelligently. This creates a complex value system for states and actions. The values are used to select our actions, and we experience them as "qualia".
So colors and pleasures are just sensory experiences accompanied by their action-values. In the end it's all a game, and qualia is generated by the game. We're mistakenly searching for qualia in the brain, while it was all around us, in the game itself. The brain is but one part of it.
Why wouldn't it? Not everyone shares your underinformed intuitions. Predictions about light are colors, predictions about reward are pleasure, etc.
The same way that blind people do?
> Colors, smells, pains, etc. are not part of the mathematical or physical description, so you can't simply dismiss the problem of qualia by labeling it as dualism. Where do the colors, tastes, feels come from?
I'm curious: what is a concrete answer that you would find satisfying? Even if it's unrealistic, or would only work in a different universe, what sort of answer are you looking for?
Color is a stand-in for any subjective (or creature-dependant) sensation, which would include sounds and feels. Remove those and epistemology collapses.
> I'm curious: what is a concrete answer that you would find satisfying? Even if it's unrealistic, or would only work in a different universe, what sort of answer are you looking for?
I have no idea. How to explain the subjective in terms of the objective? Why does such a division exist? It's a deep metaphysical question.
Most of abstract mathematics and philosophy deal in concepts that have no color, shape, or mass. And yet we deal with them. We don't need to have physical grounding for every concept.
1. A physical phenomenon, spectrum of light in this case.
2. An entity with a retina with light perceiving cells at the right bandwidth.
3. A network that represents an invariant concept at that bandwidth without too many dependencies on object and context. However, there might be visual illusions that are hard to get rid of.
4. If the entity needs to describe this concept to others you need a(nother) network that maps the concept to a sequence of utterances that maps to vocal cords or muscle movements for a keyboard.
5. If the entity has to be able to reason about the concept it needs to be able to store it for longer times and have flexible ways to combine it with other concepts.
6. If the entity needs retrospection of this behavior it needs to have a representation of itself.
I hereby directly challenge your subjectivity and qualia. It is most absolutely not an empirical fact.
Of course, you only have to have a virtual object thrown at your head in VR to discover that your brain will mistake computational processes for physical ones rather easily.
The author does not ignore that but states his assumption on the first page:
The foundation of my argument is the assumption that there exists an external physical reality independent of us humans.
EDIT: Clarification: "external physical reality" means to me that we can rule out subjectivity here.
Here's the one other part where he references subjectivity/qualia:
> In this example, the frog itself must consist of a thick bundle of pasta whose highly complex structure corresponds to particles that store and process information in a way that gives rise to the familiar sensation of self-awareness.
This is too big an big assumption to make: processing something gives rise to sensations? Just because the processing is mathematically describable, is the sensation also? What would the mathematical description of the experience of a fever chill look like? This is the typical omission made by the many people who have formed similar arguments to Tegmark.
Do we? Were? I don't see that in the essay.
> This is the typical omission made by the many people who have formed similar arguments to Tegmark.
Do you have any links? I'd like to read more.
Actually, I think you are correct—I misread part of it the first time. He is pretty meticulous about saying this only applies to 'external physical reality,' and the only reference to an internal part is in the excerpt I gave above about the sensation of self-awareness.
However, it's a similar kind of trick. He goes ahead and makes statements like:
> Everything in our world is purely mathematical — including you.
—which isn't really so significant if you are implicitly ignoring any interior aspects.
Instead, it's just kind of a tautology: the 'exterior' aspects of reality (well, the selection of them which we know of/discuss anyway) are precisely the ones approachable to conceptual/linguistic/mathematical formulation. So I don't see much more being said here than "the mathematically describable parts of reality are mathematically describable."
> Do you have any links? I'd like to read more.
I've already spent far too much time on this thread, but if you look around at adherents of digital physics, computational universe, etc. you'll find a similar elision repeating itself: either subjectivity/qualia are ignored completely, or it is assumed that they would be a by-product of some kind of generative mathematical/computational process. I've seen several articles related to the idea just on HN.
Edit: also: this is just a recent manifestation of the idea, which has been around at least since Plato.
> —which isn't really so significant if you are implicitly ignoring any interior aspects.
You are quoting the quite catchy climax of the introduction. The paragraph this quote ends with starts with "So here is the crux of my argument." The author is trying to pave the way for the actual argument and a context-free quote of the buildup does not help the discourse.
>> Do you have any links? I'd like to read more.
> I've already spent far too much time on this thread
I am sorry to have wasted your time with requests for foundations of your claims.
> Edit: also: this is just a recent manifestation of the idea, which has been around at least since Plato.
Which aspect? I actually find it very refreshing to read an essay that goes down to foundations of philosophy. If your valuable time allows, could you maybe find the time to explicitly point out congruent arguments in philosophy "since Plato"?
The reason I quoted it is to show that he makes assertions meant to sound as if they include interior aspects of reality, even though he says previously in the article that he is only talking about external aspects. There is nothing that happens in that paragraph which affects that fact at all.
To be more clear, the pattern is like: Let us redefine the universe to just refer to exterior aspects of the universe. Then for the rest of the article he can just talk about anything and the reader is supposed to call to mind the initial disclaimer. Statements like "Everything in our world is purely mathematical — including you" gain their force from the fact that he stated his limitation only early on, so that 'everything' has been redefined to mean something else.
> I am sorry to have wasted your time with requests for foundations of your claims.
If I were deferring justification that would be another matter. Instead, I was declining to give yet more elaboration of which there's plenty in this thread already.
> If your valuable time allows, could you maybe find the time to explicitly point out congruent arguments in philosophy "since Plato"?
https://en.wikipedia.org/wiki/Theory_of_forms
The common thread is that folks since Plato have tried to make arguments that general categories of things are more real than particular things themselves. The means of going about this have been various. The author's particular tack is called 'mathematical platonism': https://plato.stanford.edu/entries/platonism-mathematics/
In the prologue of "Vita Activa" she wrote this:
> This future man, whom the scientists tell us they will produce in no more than a hundred years, seems to be possessed by a rebellion against human existence as it has been given, a free gift from nowhere (secularly speaking), which he wishes to exchange, as it were, for something he has made himself. There is no reason to doubt our abilities to accomplish such an exchange, just as there is no reason to doubt our present ability to destroy all organic life on earth. The question is only whether we wish to use our new scientific and technical knowledge in this direction, and this question cannot be decided by scientific means; it is a political question of the first order and therefore can hardly be left to the decision of professional scientists or professional politicians.
> While such possibilities still may lie in a distant future, the first boomerang effects of science's great triumphs have made themselves felt in a crisis within the natural sciences themselves. The trouble concerns the fact that the "truths" of the modern scientific world view, though they can be demonstrated in mathematical formulas and proved technologically, will no longer lend themselves to normal expression in speech and thought. The moment these "truths" are spoken of conceptually and coherently, the resulting statements will be "not perhaps as meaningless as a 'triangular circle,' but much more so than a 'winged lion' " (Erwin Schrodinger). We do not yet know whether this situation is final. But it could be that we, who are earth-bound creatures and have begun to act as though we were dwellers of the universe, will for- ever be unable to understand, that is, to think and speak about the things which nevertheless we are able to do. In this case, it would be as though our brain, which constitutes the physical, material condition of our thoughts, were unable to follow what we do, so that from now on we would indeed need artificial machines to do our thinking and speaking. If it should turn out to be true that knowledge (in the modern sense of know-how) and thought have parted company for good, then we would indeed become the helpless slaves, not so much of our machines as of our know-how, thoughtless creatures at the mercy of every gadget which is technically possible, no matter how murderous it is.
> However, even apart from these last and yet uncertain consequences, the situation created by the sciences is of great political significance. Wherever the relevance of speech is at stake, matters become political by definition, for speech is what makes man a political being. If we would follow the advice, so frequently urged upon us, to adjust our cultural attitudes to the present status of scientific achievement, we would in all earnest adopt a way of life in which speech is no longer meaningful. For the sciences today have been forced to adopt a "language" of mathematical symbols which, though it was originally meant only as an abbreviation for spoken statements, now contains statements that in no way can be translated back into speech. The reason why it may be wise to distrust the political judgment of scientists qua scientists is not primarily their lack of "character" — that they did not refuse to develop atomic weapons — or their naivete — that they did not understand that once these weapons were developed they would be the last to be consulted about their use — but precisely the fact that they move in a world where speech has lost its power. And whatever men do or know or experience can make sense only to the extent that it can be spoken about. There may be truths beyond speech, and they may be of great relevance to man in the singular, that is, to man in so far as he is not a political being, whatever else he may be. Men in the plural, that is, men in so far as they live and move and act in this world, can experience meaningfulness only because they can talk with and make sense to each other and to themselves.
As for the paper:
"It is important to remember, however, that it is we humans who create these concepts" vs. "Modern mathematics is the formal study of structures that can be defined in a purely abstract way."
What's the definition of "to define"? Create a concept of something?
"the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures — we discover them"
That doesn't follow at all. What's worse, you could also use an example sentence about a child and her mother translated into various languages, and say the actual nouns don't matter, just the relation between them. That's how you discover biological relations, that's how you discover what a priest and their function is.
Last, but never least: http://www.bartleby.com/370/55.html
The question of whether it's a workable perspective or not is way above my pay grade. Some call it differential ontology http://www.iep.utm.edu/diff-ont/
The joke about category theorists being unable to distinguish isomorphic objects seems relevant (saying that as an appreciator of category theory). Also relevant is the hoary question of the identity of indiscernibles. If every distinguishable entity must stand in a unique relation to the universe, couldn't that solve the apparent ontological problem?
Whenever I try to cut you open and get at it, you keep yelling about qualia and human-rights violations.
I think there has been some miscommunication if what you got from my comment is an insistence that I (and others) are less than what's suggested in the article. What I'm speaking of is strictly addition. If anything, the ghost metaphor what be more applicable to someone claiming 'we are only' abstraction X (e.g. mathematics). Or Maybe I'm not following your intention there.
That's probably because without a proper cognitive system, you don't have intentionality ;-).
I would like to (gently) challange the assertion that all untestable hypotheses are "peers" of (neo)Platonic notions. What sets mathematics apart is this: https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...
> imagine some complex 3d object changing in time. Now imagine it's wrapped in a kind of net, like you see in 'wireframes' in computer graphics. If that complex 3D object is our universe, I see math as like that wrapping wireframe: there is a strong correspondence of some kind, and it covers the full extent of what's out there in some sense, but there are still gaps and extents into other directions etc. which are not math (or any form of human description).
I agree with you though, that it should be set apart from many more absurd hypotheses because of the additional merit it has—although for the specific question of whether reality is literally comprised of mathematics (rather than aspects of it being accurately describable by mathematics), it's a binary question, true or false, and while in some sense it's closer than the others, as far as I can tell it still lacks justification.
Edit: I should also add that my jab about probabilities approaching zero etc. etc. was more kind of a joke/flourish which probably could've been left out :)
So shut up. And calculate.
It's not math that generates qualia, it's embodiment. The fact that "math" is inside an agent, which is inside a world that evolves. The agent can move about and affect the world, and it has a utility principle (some needs to fulfill) that guide its actions and perceptions - that generates qualia. The source is being in the world, not math. Math describes anything, qualia describes the world as experienced by the agent.
If you take "being in the world, maximizing utility" as an explanation for consciousness, then you can easily classify various AIs and entities as conscious or not-conscious.
I yet to have read this essay, but I had two thoughts:
1. Similar to Turing completeness, actual physical reality might not matter for the internal observer. All Turing-complete machines can calculate same things and observer from the inside of the machine cannot determine on what hardware the machine is implemented. Similarly, this can be case for observers within our universe (humans). So maybe there is, at a high enough level of abstraction, only one mathematically possible universe.
2. The mathematical theorems are shortcuts to computation. For example, I can add two numbers on a computer, or I can use a theorem, which will tell me only some property of the result, but avoids having to do the actual calculation. So if universe is just a computation, one could avoid actually computing it in certain cases by having a theorem that would tell them enough information they need. In other words, you don't actually need to simulate anything in order for objects in simulation to "exist". So there is no need for actual hardware on which the universe runs to exist.
Addendum: So I read the essay, and I disagree that the multiverse is necessary. As I already stated in point 1, it can be the case that all possible mathematical realities are the same at high enough level of abstraction.
ftp://ftp.idsia.ch/pub/juergen/zuserechnenderraum.pdf
http://www.mathrix.org/zenil/ZuseCalculatingSpace-GermanZeni...
Thanks for the link.
EDIT: Nope guess not now I'm knee deep in this thing :P
So this theorem gives you a property of the result (the result is larger than some number) without you having to calculate the result (sum the numbers) explicitly.
The Hashlife algorithm mentioned in this thread is another case, basically it's an algorithm to prove certain "theorems" about the Game of Life and applying those theorems to speed it up.
Read his book 'Our mathematical Universe', he explains there why these are actual separate multiverses.
I agree it's somewhat a matter of interpretation whether it's truly "multi" or a single entity. But if the domains are separate enough, it's probably best to think of them as different universes.
You may enjoy the science fiction book Permutation City
- What he describes in the abstract is nothing new. Part of your work as an applied mathematician is to take mathematical models developed by others, e.g, in the form of systems of differential equations, and treat them as absolute truths (for the sake of computation; essentially you create "a mathematical bubble" for yourself and choose to live in it), study the consequences, and present the results. So his claim of "I advocate" is disgusting plagiarism/credit-snatching of a well-known technique.
- What he actually advocated is total bullshit as far as physics is concerned. You can live as long as you want in the bubble of your mathematical models. But if you have to connect your math to reality, you have to make your math subordinate to empirical data, and if the data disagrees, your shiny mathematical model is wrong, period. Also don't forget what George Box said "All models are wrong. But some are useful".
(My favorite: https://en.wikipedia.org/wiki/One-electron_universe)
I do not say that to belittle physicists. Not at all, I have the utmost respect for what they do.
But as they dig deeper and deeper, what they come up with sounds like the universe is a pretty strange place to be. Personally, I am okay with that idea, because, if I look closely enough, I have been living all my life in a world that does not really make sense. But it is really strange, nevertheless.
The halting problem (endless loops) is sidestepped by being contained to a single universe. Also, apparently our universe got lucky as we get to live in an ordered universe, and not dissolve into nothingness due to a bug in our universe's binary code.
Of course, like "normal" mathematics, "reality" mathematics should also be incomplete. Furthermore, there is no way to know everything of the universe one is a part of, as this requires at least 1 bit more than there are available bits in the universe. But this still allows for the possibility of the universe itself being an inference machine.
Edit: Then again, I believe that mathematics is a mental construct, mathematics does not exist without brains, or if it really exists external to human minds, it exists in all possible forms: Humans do not discover maths as a whole, they cast a local net of subjective observation over it, and keep the things that make sense to our brains and are consistent within the current framework. Again, this does not preclude the possibility that our mathematical reality exists inside the mind of a giant supercomputer.
>I believe that mathematics is a mental construct, mathematics does not exist without brains
Brains evolved through natural selection to survive in the universe we exist in. Ideas similarly "evolved" to be good at modelling the universe. If mathematics didn't exist outside of our head, it would be useless. If there are two apples on the ground, and two more fall of a tree, then there are four apples on the ground. This is true whether or not a human is there to observe it and count them.
I don't know about that. How close do the apples have to be to number them? If two apples are within inches and another a mile away, is there two apples or three apples? Define ground. Is it dirt? And dirt of what? If two apples are on Earth and one apple on Mars, is there two or three apples on the ground? What does it even mean to number things? What is a number?
My point is that it takes a human brain for establishing definitions, axioms, and conditions. Only after these have been established can we do math.
There are three possible answers to this: The binary string itself is a form of computation, with 1's being upwards fluctuation and 0's being downwards fluctuations. Something like a Boltzmann Brain. Second, Zuse-style, the universe itself could be a computer: A Turing-complete cellular automata, acting on the input formed by a random process. Third, our universe may be mental and the binary forms mental states.
> This is true whether or not a human is there to observe it and count them.
This may be true, only because human brains made it true. Objects, like apples, or apple trees, are somewhat arbitrary selections of parts of the universe (The human chose to wall off a certain subset of everything).
With a little imagination you could see the apple on the ground and the tree it fell from as one object. Why is a hair on my head part of the object "me", but the moment I pull it out, it becomes its own object? What if the apple on the ground rots away and turns to mush? Is it still its own countable object? What if a cow eats the four apples? Is there now no object, one object, or still four objects?
Your view is much closer to reasonable than mine though, because to prove that maths is a mental construct, I have to invoke scientific unpopular views, like intuitionism (maths is an advanced form of Wittgenstein's language games) or Indian/Asian mysticism (make me one with everything).
But the reason that intuitionism is unpopular, is not because we found it to be objectively untrue, but because of the very human element of Hilbert hating Brouwer's guts. If Brouwer had more clout then we may see the Law of excluded middle as untrue or unprovable, as it stands, we entered Hilbert's timeline where this law is deemed objectively true even outside of minds.
Then again, my view has no problem with "Spooky action at a distance", because I am not required to view two electrons as separate objects, and distance between arbitrary subsets is relative (owes its value) to the chosen metric, location and measurement equipment of observers.
If the sun rises I can capture this with maths and prove to you that the sun is not going down. But I forgot to add "the sun rises to me", I forgot the creative power of observation: Because for someone on the other side of earth the sun is going down. For someone in space the sun may not appear to move at all.
[1] https://en.wikipedia.org/wiki/Multiverse#Max_Tegmark.27s_fou...
Her more recent article (review of Max Tegmark's book) also comments on the same subject: http://backreaction.blogspot.in/2017/11/book-review-max-tegm...
If you really consider the issue closely, you'll see it depends on not overlooking certain dark corners of it which you can sort of ignore and assume to not contain contradictions—but they're there! And once you see them, it's hard to unsee them: what in the world would it mean for a desk to 'made' of mathematics—it's nonsensical. I think programmers are especially susceptible to this mistake because they're used to using mathematical descriptions to generate things, but even there, the idea that the math is doing any generating is a bias of our perspective since we're focused on the math; what's actually going on is a physical process involving silicon and plastic and metals etc., which can also be described mathematically, but the leap to being math is an unfounded step which could probably be predicted by the fact that that's where our focus lies.
It also begins to look kind of silly when you consider how human math is: it's an outgrowth of an evolved mental faculty. Cognition is certainly impressive relative to the other things we and various animals are able to do—but it's probably still in the same category of our other faculties like the olfactory, visual, tactile etc.—and unlikely to be involved in the literal fabrication of the universe.
> I advocate an extreme “shut-up-and-calculate” approach to physics, where our external physical reality is assumed to be purely mathematical.
> Here, I will push this idea to its extreme and argue that our universe is not just described by mathematics — it is mathematics.
> Everything in our world is purely mathematical — including you.
Here's one mistake he makes: he doesn't account for qualia, the fact that people have subjectivity. Here is the one sentence where he glosses over it:
> In this example, the frog itself must consist of a thick bundle of pasta whose highly complex structure corresponds to particles that store and process information in a way that gives rise to the familiar sensation of self-awareness.
This is a pretty big assumption to make: processing something gives rise to sensations? Just because the processing is mathematically describable, is the sensation also? What would the mathematical description of the experience of a fever chill look like? And what about the 'system' that carries out the transition from process to sensation? Could it even be called a 'system' or is it more on the order of the fever chill itself?
Edit: I clarified a few things a couple minutes after posting.
> I'm don't think that is a sufficient argument to explain subjective experience,
—I have to agree with that, and I haven't seen anything which makes me think a better justification might be forthcoming (from anyone, not just Tegmark), so I kind of stop there.
My personal view on it is that qualia/subjectivity are just what it's like to actually be a thing ('on the inside' is kind of an approximate way of saying it). Considering the sorts of things that go on in human subjectivity (this time referring to objects appearing in it, not the medium itself), I suppose it is what it's like to be the part of the reality where a human brain is (actually I think it's only certain parts of human brains, probably related to executive function). In that case, I think a bunch of logic cates linked together in certain ways could end up having interesting 'internal aspects' (of course 'interesting' here is human-relative: who's to say the the parts of reality comprising trees aren't internally more interesting in some objective measure? (not that I think there is an objective measure for that.)
But I qualify the above as just how I guess that things are, not an assertion that it's really the case or anything.
What does it really mean to "exist" at all? It's equally undefined. This question is either problematic for every metaphysics, or none of them.
A desk is not "made" of mathematics any more than the words you're reading right now are "made" of electrons. What matters for existence is the relationships between mathematical structures. In this case, the relationship of your mind perceiving a desk is identical regardless of whether your mind and the desk are mathematical structures or some non-mathematical matter.
And once you accept that isomorphism, it's an easy step to a type of Platonism/mathematical monism because it's so unproblematic as a philosophy of mathematics. And once you're there, you have the mathematical universe.
Well, that's what Structuralism is (https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_s... —but it would be running in circles to say that Structuralism is the correct approach here since it would only be the correct approach if the universe is in fact made of mathematics (in other words, it would be assuming your conclusion in your argument).
To be convincing, it's not necessary to claim that any particular position is definitely true, I should only need to demonstrate that said position solves open problems that have not or cannot be solved by other positions while being of comparable complexity, conceptually.
This is indeed the case for something like the mathematical universe over naturalism. There may yet be a comprehensive, unproblematic account for mathematics in naturalism, but we certainly don't have it yet. That's a huge, gaping wound in our fundamental knowledge.
The mathematical universe, by contrast, easily grounds mathematics and matter by encompassing both in the former.
The philosophical implications are far broader than that. The mathematical universe provides working foundations for a lot of philosophy and physics. For instance, the multiverse entailed by the mathematical universe explains why fine tuning for life isn't necessary.
> but I'll point out it's better not to pretend we have an answer if we really don't--then we keep looking.
You can keep looking despite accepting that the mathematical universe is currently our best explanation. Your implication that we wouldn't do this is simply bizarre. Would you shout equally loudly that we shouldn't build on our best scientific theory simply because it may not explain everything?
> The idea that reality is made of math as an answer is totally lacking in justification
Except I just covered the justification: the fact that it explains plenty of philosophical and scientific problems without introducing unnecessary complexity. What more justification could you possibly need?
Frankly, it sounds to me like you're either special pleading or moving the goalposts simply because you don't like the idea of the mathematical universe.
By the way, if you enjoy reading this piece, you might also enjoy really interesting paper coauthored by the same person: "Why does deep and cheap learning work so well?," available at https://arxiv.org/abs/1608.08225 -- it posits that deep learning works well because of the particular structure of the laws of physics in the universe in which we happen to live.
Hasn't it been understood for a long time that quite a bit of things were entirely non-deterministic ? eg radioactive decay. You could compute a possible state, but the universe is a sequence of state spaces.
So then you're saying reality can't have a logical contradiction? Seems like you're agreeing with the original statement.
>Even a totally random universe follows some kind of distribution
Which is just a kind of rule.
He's assuming a mathematical model that faithfully represents reality, hence "reality can't contain contradictions". We're not going to one day find a contradiction and vanish in a puff of logic, the ultimate model of reality will be contradiction-free.
>So reality works because math works.
Reality is that the one that contains the structures not math. So what you said does not make sense.
But math contains all physically realizable structure, i.e. all structure that can represent some feature of reality. But also structure that cannot be realized in reality, e.g. sets too big to fit in reality. That one can deduce some logical structure from observing reality isn't the deciding factor here.
I think, Tegmark does not talk about computations per se because the hypothesis is agnostic about what kind of computer our universe is. It could, for example, be a geometric computer in which the position of objects can be determined with infinite precision (i.e. using the real numbers). Such a computer would be strictly more powerful than a Turing machine or equivalent (i.e. all computational models that can be described and run inside a Turing machine). Enumerating the space of all formalisms (i.e. the domain of mathematics) is as agnostic as you can be.
That is, to me, a weird mixing of scales. The universe only seems complex to us because of the scale at which we experience it in terms of time and physical extent as well as the convoluted way our own systems of understanding developed and then were later mutated and stretched into different forms to be better suited, etc. By definition the universe itself can't really be more or less complex than anything. At least not anything we can experience.
The idea of a geometric computer is surprisingly close to things I've been considering recently, but 'position of objects can be determined with infinite precision' seems strange. It may be just that I don't understand what is fully meant when you say 'geometric computer' but I would expect 'position' would be mostly meaningless in that space except as an emergent/illusory 'property'... and I don't think arbitrary precision would necessarily be implied either.
The question, in my mind at least, is how to show that the quantum understanding leads, on the macro scale, to the emergent phenomena we observe and once thought unitary. That's a great degree of precision and division between entities to arise out of the nonlinear interactions of bunches of 'particles'. And given the scale on which such things emerged (the universal one), clearly a proper system of understanding would make the emergence of things like atoms, molecules, chemistry, astronomy, geologies, planets, solar systems, galaxies, stars, and all those things self-evident. But... save for a few exceptions (more all the time though which is encouraging!) we rarely even look at things en-masse. It's all well and good to know how 2 bodies interact under gravitation. But when your system falls apart with 3, and you really need to be able to toss in a trillion with dozens of other nonlinear relationships... It really ought to be more obvious, I would think, that we're missing something.
IIRC there are no arguments about computation or what underlying medium(s) are involved, it seems like a separate question.
These are actually related by Goedel's incompleteness theorems and the Curry-Howard isomorphism. Computation and mathematics are inextricably linked.
The Computable Universe is a necessary restriction on the mathematical universe hypothesis to avoid accepting inconsistent universes.
If we all just shut up and calculated, we'd still be doing Lorentz transforms near speed of light "just because they work", right?
The whole point of that catchphrase "shut up and calculate!" was supposed to be "stop speculating about things we can't prove or disprove and wouldn't have any direct impact on us if we could; focus on crunching the numbers and figuring out what's effective."
This whole "mathematical universe" thing seems like practically the definition of premature speculation! It's no more useful than ancient Greek philosophers speculating on the nature of the elements. Those other people over there drawing circles in the sand, now, they're getting somewhere.
A universe with true randomness might satisfy that literally. But random variables can be incorporated into a mathematical model. But what would it even mean for a universe to not be describable by any kind of math at all? I don't think such a concept is coherent or discussable.
> [Model-agnosticism] consists of never regarding any model or map of the universe with total 100% belief or total 100% denial. Following Korzybski, I put things in probabilities, not absolutes... My only originality lies in applying this zetetic attitude outside the hardest of the hard sciences, physics, to softer sciences and then to non-sciences like politics, ideology, jury verdicts and, of course, conspiracy theory. -- RAW
A universe isn't a map, it is the territory. An ordered universe can still be mapped/modeled with maths. A universe that is impossible to be mapped/modeled and has no predictable patterns would be an impossibly boring universe indeed (Kolmogorov Random Universe). Perhaps our brains would evolve to think more in probabilities than absolutes in such a universe. Or they could simply not be allowed to exist (intelligence requires order).
While we may not be able to predict the location of a single molecule of air in a mere 10,000 years with a rough accuracy of a cubic femtometer, the universe does it. And our inability to predict it is not because we need a bigger computer. Our inability to predict it is because in order to predict it, we would have to simulate the entire universe and every single particle within it, and every interaction between all of them, to infinite (ehhh... maybe) precision. Which would have an information load exactly as big as the universe itself and increase entropy (which implies it takes as much time) as much as actual evolution of the entire universe over that course of time. To predict one molecule of air. And the universe does it for all the molecules of air. Along with all the other particles.
But it does not.
It does not do that, either. (Because there is no "end result" to calculate.)
It's not so much that chaotic systems can not be calculated but instead that enough precision of starting conditions can't be obtained to ensure an accurate result (for example, highly non-linear systems). The double pendulum system is a perfect example where computation is simple yet the system is still chaotic.
Of course, you can't write down the real-number result with infinite precision, can you? You'll have to chop it off at some point. That's OK. That's what error bars are for. Just keep going and keep track of your error. Oops, you'll find that your error grows larger than the prediction itself with remarkable speed! THAT is the chaos. It doesn't matter if you measured the starting conditions with literal infinite perfect precision. The error bars on your predictions are going to explode like a busy beaver function. You're not going to catch that. Nothing can. (I imagine the actual physical limit would be that your predictions can be accurate within a distance equivalent to the radius of a sphere expanding at the speed of light from the initial position but I haven't actually read anything suggesting that, it just seems sensible.)
Then you've got stupid simple cellular automata like rule 31 1D systems. Totally discrete. Totally basic. No precision even needed anywhere. And yet, you can't predict it. Not without accumulating an error greater than the state in the system itself if you skip even the slightest act of computation necessary to evolve the system as a whole from one step to the next.
If this were true, it should be trivial to prove by observation that we're not in a mathematical universe, or that physics is not computable, even in principle. As far as I understand, physics as we know it is widely acknowledged to be computable, albeit with exponential slowdown on classical computers.
By the Bekenstein bound and other thermodynamic constraints, a finite region of space necessarily contains a finite amount of information, or it would collapse into a black hole, ergo physics is at worst a finite state machine. So where's the disconnect?
Edit: Hm, actually, the lecture was about dangers of AI, so not the same topic.
> When we derive the consequences of a theory, we introduce new concepts — protons, molecules, stars — because they are convenient
Couldn't matter itself be just another "human", "convenient" concept? Indeed the author writes:
> our external physical reality is assumed to be purely mathematical
Why should a purely mathematical reality also be material? Doesn't this lead to the logical conclusion that ideas, err, maths is "all that there is"?
Sure, in this case ideas are objective and not subjective, but Berkeley's "mind of God" argument also leads to an objective, albeit immaterial reality, doesn't it?
2) Materialism and "objectivism" are closely tied together. Mathematics consists of abstractions (or models), and those reside, in particular, in our mind; being "convenient", these abstractions are therefore reflections of what lies outside, and so - almost by definition - the objective world itself, matter, cannot possibly be built from math. (Besides our mind, "abstractions" can also reside in other, inanimate, objects; regardless, this still means that some part of matter merely reflects, to a degree, some other part).
No, it leads to a subjective inability to know anything at all. It's a difficult philosophical view to talk about, because the people espousing it can not possibly actually believe it and usually don't even understand it. The non-existence of a shared, knowable, objective reality means that, even in a totally solipsistic model, inductive reasoning can not be applied. You can not assume that attempting to breathe in the next moment will extend your life rather than extinguish it. You can not conclude that the air in your lungs will not suddenly become ants. You certainly can not leap to the extensively complex chain of reasoning necessary to believe that between this sentence and the next the English languages will turn into French. It introduces aggressive, pervasive, impossible-to-dismiss inconstancy in all things, including ones self.
A true believer in a subjective universe would remain still in their bed, doing nothing, until they died. Any action whatever beyond automatic biological functions instantly betrays their belief that they know their body, know the environment they are in, and know how the mental activity necessary to cause the action will elicit a change in the environment which is not likely to destroy them. And that is a thing they can never know in a subjective universe.
The 'mathematical universe' on the other hand is simply discussing how the shared, knowable, material universe we live in came to be or is best understood. I happen to think it is factually wrong on a few points (the author just off-hand claims that if we had a big enough supercomputer we could exactly calculate the future development of the universe which is fundamentally untrue for one) but it certainly doesn't amount to subjectivity.
Subjective existence cannot be reduced to Math because it is our explorations in subjectivity that led to math’s invention and refinement
It is precisely whether Maths is a subset of subjective experience that we invented or whether it has always been a superset of the universe our subjective experience is embedded in that we merely discovered is precisely what is at issue here.
It's easy to get confused by the language used to frame things. I could say that language only has countably infinite possible statements, so it must therefore be impossible to use set theory which is learnt through and derives from language to analyse uncountably infinite sets, but it would be a bad argument.
I don't think that subjectivity can be reduced to math, but reducing math to subjectivity doesn't obviously imply that subjectivity cannot be reduced to math, as far as I can see right now.
I am not completely certain in the subtleties of non-standard set theory, but at least if we replace the axiom of Choice (AC) by the axiom of Determinacy (AD) - which is perfectly fine, though it is less common to do mathematics in this system - we can partition the real line R by an equivalence relation that has more equivalence classes than elements of R. So at least we can partition R into more disjoint subsets than R has elements.
Source: https://math.stackexchange.com/questions/1104028/what-are-di...
That's not surprising, since an equivalence relation is a function of two elements of the set, i.e. O(N^2), and a subset is a function of a single element, i.e. O(N) . The OP mentioned subsets, and an equivalence relation is not a subset of the original set--it doesn't even typecheck.
I don't understand your reasoning. Every equivalence relation partitions the set S on which it is defined on into subsets - these are called equivalence classes. Every equivalence class contains at least one element of S. So I find it quite surprising that there are more equivalence classes than elements of S (in this case S are the real numbers) if we assume AD.