Follow-up to: Syntacticism

I wrote:

The only objects that are real (in a Platonic sense) are formal systems (or rather, syntaxes). That is to say, my ontology is the set of formal systems. (This is not incompatible with the apparent reality of a physical universe).

In my experience, most people default^{1} to naïve physical realism: the belief that "matter and energy and stuff exist, and they follow the laws of physics". This view has two problems: how do you know stuff exists, and what makes it follow those laws?

To the first - one might point at a rock, and say "Look at that rock; see how it *exists* at me." But then we are relying on sensory experience; suppose the simulation hypothesis were true, then that sensory experience would be unchanged, but the rock wouldn't really *exist*, would it? Suppose instead that we are being simulated *twice*, on two different computers. Does the rock exist *twice as much*? Suppose that there are actually two copies of the Universe, physically existing. Is there any way this could in principle be distinguished from the case where only one copy exists? No; a manifest physical reality is observationally equivalent to N manifest physical realities, as well as to a single simulation or indeed N simulations. (This remains true if we set N=0.)

So a true description requires that the idea of *instantiation* should drop out of the model; we need to think in a way that treats all the above cases as identical, that justifiably puts them all in the same bucket. This we can do if we claim that *that-which-exists* is precisely the *mathematical structure* defining the physical laws and the index of our particular initial conditions (in a non-relativistic quantum universe that would be the Schrödinger equation and some particular wavefunction). Doing so then solves not only the first problem of naïve physical realism, but the second also, since trivially *solutions to those laws* must follow those laws.

But then why should we privilege our particular set of physical laws, when that too is just a source of indexical uncertainty? So we conclude that all possible mathematical structures have Platonic existence; there is no little XML tag attached to the mathematics of our own universe that states "this one exists, is physically manifest, is instantiated", and in this view of things such a tag is obviously superfluous; instantiation has dropped out of our model.

When an agent in universe-defined-by-structure-A simulates, or models, or thinks-about, universe-defined-by-structure-B, they do not 'cause universe B to come into existence'; there is no refcount attached to each structure, to tell the Grand Multiversal Garbage Collection Routine whether that structure is still needed. An agent in A simulating B is not a causal relation from A to B; instead it is a causal relation from B to A! B defines the fact-of-the-matter as to what the result of B's laws is, and the agent in A will (barring cosmic rays flipping bits) get the result defined by B.^{2}

So we are left with a Platonically existing multiverse of mathematical structures and solutions thereto, which can contain conscious agents to whom there will be every *appearance* of a manifest instantiated physical reality, yet no such physical reality exists. In the terminology of Max Tegmark (The Mathematical Universe) this position is the acceptance of the MUH but the rejection of the ERH (although the Mathematical Universe is an external reality, it's not an external *physical* reality).

Reducing all of applied mathematics and theoretical physics to a syntactic formal system is left as an exercise for the reader.

^{1}That is, when people who haven't thought about such things before do so for the first time, this is usually the first idea that suggests itself.

^{2}I haven't yet worked out what happens if a closed loop forms, but I think we can pull the same trick that turns formalism into syntacticism; or possibly, consider the whole system as a single mathematical structure which may have several stable states (indexical uncertainty) or no stable states (which I think can be resolved by 'loop unfolding', a process similar to that which turns the complex plane into a Riemann surface - but now I'm getting beyond the size of digression that fits in a footnote; a mathematical theory of causal relations between structures needs at least its own post, and at most its own *field*, to be worked out properly).

If you notice that "exists" no longer makes total sense, the way forward is not to just throw intuition at the wall and see what sticks. The way forward is to stop talking about "exists" for a while.

Yes, but I'm actually going somewhere with this: EY doesn't believe that "infinite sets exist" (loosely put). So I'm trying to deduce constraints on what "exist" can mean and still be coherent; at this point I don't think we can even claim that lavalamp's kangaroos "don't exist".

So if you've ever read Probability Theory, by E.T. Jaynes, I suspect he's just a convert to the position that, in order to make sense when applied to the real world, infinite things have to behave like limits of finite things. Using "exists" can be avoided.

I haven't; I probably should.

Is this "limits" in the sense of analysis (epsi-delta limits), or is it "limit points" (like ω)? If the former, then that position involves not believing that arithmetic makes sense when applied to the real world. If the latter, then the position doesn't seem different from what most mathematicians believe, because allowing limit points gets you transfinite induction... but then, as I intend to show, that gets you the first uncountable ordinal, by the power of 'scary dots'. So... either EY doesn't believe arithmetic applies to the real world, or EY doesn't know the logical consequences of his beliefs, or EY doesn't believe the above position, or I've made an error. Of course, at this point the last of those is rather likely, which is exactly

whyI want to formalise my argument and lay it out as coherently as possible.Also, if I can't say "exists", how come you can talk about "the real world"? Double standards if you ask me ;)

Well, you

cansay exists, it just seems to be digging you into a hole.Has EY articulated a position on "infinite set atheism" outside of a few off-hand comments, that are possibly jokes? You might be preparing to rebut a position that he hasn't taken.

Well, I've seen him go into quite some detail in comment threads about uncountable ordinals; it looks to me very much as though there is a position there that he's taking (it's fairly clear that it's more than just a joke). OTOH, I haven't seen an article by him setting out his position in a rigorous or articulated fashion, though if there is one I'd like to know about it.

No. Look, if you think rocks exist now learning spooky metaphysical things about the nature of the universe should not change that belief. 'Rock' is just an entity our model of reality uses to help us make predictions and control things. That's what it means for something to exist.

Now, lots of mathematical objects are useful entities to have in our model. So we can say they exist- of course they aren't variables in the causal model, they're isomorphs of variables and the structure in the model that help us use the model more effectively. We'll call these entities 'abstract' and the actually causal variables in the model 'concrete'.

Whether or not there are other universes where abstract objects are concrete objects seems like an open question to me and not something that can be answered with philosophical speculation.

This implies you believe that it is logically possible, so you do think philosophical speculation can contribute to seeking the answer.

I don't think there is a clear distinction between science and philosophy- so yeah it can contribute to the answer. My point is more that the question is contingent in the same way that, say, physics is contingent. If the scientific-philosophical enterprise ever outputs the Tegmark Level 4 multiverse: cool. But you don't get there through mere reflection on the meanings of words (if the history of philosophy teaches us

anything...).All I see here is Tegmark re-hashed and some assertions concerning the proper definitions of words like "real" and "existence". Taboo those, are you still saying anything?

Have you read any of Paul Almond's thoughts on the subject? Your position might be more understandable if contrasted with his.

I'm saying that our intuitive concepts of "real" and "existence" have no referents, that Tegmark's restriction to computable structures is unnecessary, that nesting (ie. simulation) of worlds is an explicit causal dependence, and that Platonism needn't be as silly and naïve as it sounds. Also to the extent that I

amrehashing Tegmark, I'm doing so in order to combine it with Syntacticism and several other prerequisites in order to build a framework that lets me talk about "the existence of infinite sets", because I think Eliezer's 'infinite set atheism' is a confusion.I'll read "Minds, Substrate, Measure and Value" (which seems relevant) and then get back to you on that one, ok?

Thanks, that's a concise and satisfying reply. I look forward to seeing where you take this.

To Minds, Substrate, Measure and Value Part 2: Extra Information About Substrate Dependence I make his Objection 9 and am not satisfied with his answer to it. I believe there is a directed graph (possibly cyclic) of mathematical structures containing simulations of other mathematical structures (where the causal relation proceeds from the simulated to the simulator), and I suspect that if we treat this graph as a Markov chain and find its invariant distribution, that this might then give us a statistical measure of the probability of being in each structure, without having to have a concept of a physical substrate which all other substrates eventually reduce to.

However, I'm not sure that any of this is essential to my OP claims; the measure I assign to structures for purposes of forecasting the future is a property of my map, not of the territory, and there needn't be a territorial measure of 'realness' attached to each structure, any more than there need be a boolean property of 'realness' attached to each structure. I note, though, that, being unable to explain why I find myself in an Everett branch in which experiments have confirmed the Born rule (even though in many worlds (without mangling) there should be a 'me' in a branch in which experiments have consistently confirmed the Equal Probabilities rule), I clearly do not have an intuitive grasp of probabilities in a possible-worlds or modal-realistic universe, so I may well be barking up the wrong giraffe.

EDIT: In part 3, Almond characterises the Strong AI Hypothesis thus:

I characterise my own position on minds thus:

This is because the idea of a 'physical system' is an attachment to physical realism which I reject in the OP.

Thanks for following up on Almond. Your statements align well with my intuition, but I admit heavy confusion on the topic.

Of course it would, though it's underlying reality wouldn't be what we expect. (it might be a pointer in some program, instead of a collection of atoms)

Unless we're talking about amplitude of everett branches I don't know what it means for something to "exist" more. As far as I know from any given perspective, existence is a binary quantity, something either is or isn't.

I don't see why. In Object-Oriented Programming, there's Classes which are a description of the instantiated objects, and there are the objects themselves, which you can do different things with than with the classes.

Again, how do you know there's no XML tag attached to the mathematics of our universe?

You seem to me to be arguing in circles, stating as evidence that which you desire to prove. The Tegmark IV hypothesis is exciting but I don't see how it's supposedly so self-proving as its proponents suggest. That seems to me a huge map-territory confusion... Perhaps map and territory

areindeed inexorably linked at the deepest levels of reality, but this can't be proven by just claiming it...Good objections, but I think they all have answers:

But a pointer is

information. At the physical layer, it's aconfiguration- when you say char *p=malloc(1); you haven't created an electron to sit in &p, rather you've configured existing electrons so as to have a pattern that is meaningful to one of your higher-level maps, but as a good reductionist you know that your higher maps don't represent anything ontologically basic about the territory.That's the point - I'm showing that the belief that 'existence' results from physical manifestation reduces to the absurdity that more physical manifestations of something means it exists more, which is nonsensical.

Does this make String("5")

exist twice as hard? No; it doesn't even change whether it exists or not. It may be an object rather than a class, but it's still just acollection of information, which can be represented once, many times, or not at all, without ever changing the value of String("5").toInteger().I don't

knowit, but such a tag would be epiphenomenal, and therefore it makes sense to assume it doesn't exist. If you're so sure that physical reality is manifest/instantiated, thenhow do you know, andwhy does it happen to follow regular mathematical laws? I think that believing in manifest physical reality is like believing in p-zombies (or, which comes to the same thing, dualistic consciousness).It's not self-proving - at least, I don't claim it so - it's just simpler (shorter message length) to say "{equations}" than to say "mass-energy and space and time, which follow {equations}", and thus in the absence of evidence for the existence of manifest reality (which evidence I do not believe to be possible) a Bayesian, taking an Occamian prior, should conclude that the former is more likely than the latter. The only reason we mostly believe the latter is that, like Copenhagen, it came first.

EDIT: also, my views are distinct from (and were formed before I read) Tegmark IV; in particular his paper makes no mention of the concept of instantiation (and seems to me to be assuming that

allTegmark worlds are physically instantiated), and he wants to limit his hypothesis to computable structures, which I consider unnecessary and unjustified - I'm happy with the existence of structures so transfinite we can't even imagine them, perhaps because I'm a mathematician rather than a physicist.I know that -- because as a good reductionist I recognize that my map is a

differentportion of the territory, that portion of the territory which resides inside my head and which is merelycorrelatedto the actual territory outside of my head.But tsn't that the opposite of what you and Tegmark IV are saying: namely that the maps we call "equations" represent something ontologically basic about the whole of the territory?

I'd be careful what I'd call "nonsensical" when your argument instead must lead to the conclusion that things must exist even when they don't exist... This would be called nonsensical by many others.

Epiphenomenalism is usually referring to something conceptual or cognitive (e.g qualia or consciousness or subjective experience) that doesn't have an influence on physical states.

You're instead talking about physical existence not having an influence on

conceptualentities (namely the mathematical equations).Yes, once you have the physical reality of a mouth and a keyboard (which happen to require mass-energy) to be able to say those equations, you can just say the "{equations}" --- namely you can transform a portion of the territory into an accurate map of the whole of the territory. But it still doesn't conclusively explain how the territory came to be.

But the actual-territory is not (or at least, need not be) causally influenced by the territory inside your head that's implementing the map.

I can't speak for Tegmark, of course, but what I'm saying is that "equations"

arethe territory, and the stuff that looks to us like rocks and trees and people and the Moon is just a map.On the contrary, the conclusion is that things must exist even when they don't "exist" - where that quotation refers to some silly little savanna-concept we have in our brains, about rocks and trees and people and the Moon. Which don't exist.

That's because (in my model) the conceptual entities are the bedrock of the hierarchy, and physical existence is strongly analogous in this model to qualia in a physical-realist model. After all, "{equations}" and "rocks following {equations}" both give the same result for "value of X at time T", so the existence of rocks is epiphenomenal to the equations.

But a simulated me, existing only as information represented by electrons in a computer, could say "equations" just as loudly. So why couldn't a purely informational me, existing as unrepresented information, say "equations" too? Physical reality is a

burdensome detailwhich doesn't add any explanatory power to your model; the claim that information needs to be represented in order for conscious entities contained within that information to exist seems to me to have no evidence backing it up, nor indeed to be capable ofhavingsuch evidence, and therefore Occam demands that we frame our model in such a way as to make that claim inexpressible. It's rather like moving from configuration space to relative configuration space; unmeasurable claims become unreal.It doesn't need to "come to be"; 'time' and 'causality' are parochial notions, concepts we can use to model things within our universe. Expecting the multiverse to obey them seems to me to be a Mind Projection Fallacy. A block universe just

is.Thanks for the insightful critique, by the way - it's helping me to understand the arguments better and see weak points that I wouldn't have noticed myself. I'm still not sure whether my theory is circular, nor whether I should care if it is.

When actually run, it makes two pieces of the territory change so that they contain a pattern that we would recognize as "5".

What does it mean for information to exist while not being represented at all, anywhere? Can you give an example of information with that property?

Right, but it doesn't attach little "" tags to

that pattern.Well, trivially not, because by giving the example I create a representation. But, does a theorem

become truewhen it is proven? That seems to me to be absurd. Counterfactually, suppose there were no minds. Would that prevent it from being true that "PA proves 2+2=4"? That also seems absurd. I can'tproveit's absurd, but that's because a rock doesn't implement modus ponens (no universally compelling arguments).If X will happen tomorrow, then it is a fact that X will happen tomorrow, even though (ignoring for now timeless physics) tomorrow "doesn't exist yet", and the information "X will happen tomorrow" needn't be represented anywhere to be true; it inheres in the state of the universe today + {equations of physics}. Information which can be arrived at by computation from other, existing information, exists - or perhaps we should move the 'other, existing information' across the turnstile: it is an existing truth that (Information which can be arrived at by computation from foo) can be arrived at by computation from foo. Tautologies are true.

I said information, you said theorem-- I don't think it's the same.

I was expecting you to say something like "the 3^^^3rd digit of pi", and then I was going to say something, but now that I think about it, I think it's time to taboo "exist".

"Theorem Foo is true in Theory T" is information. Though the 3^^^3rd digit of pi is good too; I want to hear what you have to say about it.

Ok... "exist" doesn't have a referent. Any attempt to define it will either be special pleading (my universe is

special, it"exists", becauseit's the one I live in!), or will give a definition that applies equally to all mathematical structures.I was going to say, it {can be calculated}-exists, but it does not {is extant in the territory}-exist. It certainly has a value, but we will never know what it is. No concrete instance of that information will ever be formed, at least not in this universe. (Barring new phyisics allowing vastly more computation!)

Thanks, I think that's the clearest thing you've said so far.

I think my own concept of "exist" has an implicit parameter of "in the universe" or "in the territory", so it breaks down when applied to the uni/multiverse itself (what could the multiverse possibly exist in?). Much like "what was before the big bang" is not actually a meaningful question because "before" is a time-ish word and whatever it is that we call time didn't exist before the big bang.

But then, how do you determine whether information exists-in-the-universe at all? Does the number 2 exist-in-the-universe? (I can pick up 2 pebbles, so I'm guessing 'yes'.) Does the number 3^^^3 exist-in-the-universe? Does the number N = total count of particles in the universe exist-in-the-universe? (I'm guessing 'yes', because it's represented by the universe.) Does N+1 exist-in-the-universe? (After all, I can consider {particles in the universe} union {{particles in the universe}}, with cardinality N+1) If you allow encodings other than unary, let N = largest number which can be represented using all the particles in the universe. But I can coherently talk about N+1, because I don't need to know the value of a number to do arithmetic on it (if N is even, then N+1 is odd, even though I can't represent the value of N+1). Does the set of natural numbers exist-in-the-universe? If so, I can induct - and therefore, by induction on induction itself, I claim I can perform transfinite induction (aka 'scary dots') in which case the first uncountable ordinal exists-in-the-universe, which is something I'd quite like to conclude.

So where does it stop being a heap?

I'm not saying my universe is special just because it's the one I live in, in fact I can accept the reality of lots of Everett branches in which I don't live in.

More to the point I believe that the reality of those Everett branches preexisted your mathematical models of them, or indeed the human invention of mathematics as a whole. Mathematical structure were made in imitation of the universe -- not vice versa.

Ok, now taboo your uses of "reality" and "preexisted" in the above comment, because I can't conceive of meanings of those words in which your comment makes sense.

The thing about tabooing words, is that we find it

easyto taboo words that are just confused concepts (it's easy to taboo the word 'sound' and refer to acoustical experience vs acoustic vibrations), and we find it hard to taboo words that are truly about the fundamentals of our universe, such as 'causality' or 'reality' or 'existence' or 'subjective experience'.I find it much easier to taboo the words that

youthink fundamentals -- words like 'mathematical equations', namely 'the orderly manipulations of symbols that human brains can learn to correspond to concepts in the material universe in order to predict happenings in said material universe'To put it differently: Why don't you taboo the words "mathematics" and "equations" first, and see if your argument still makes any sense

I tabooed "exist", above, by what I think it means.

Youthink 'existence' is fundamental, but you've not given me enough of a definition for me to understand your arguments that use it as an untabooable word.I'd say that (or rather 'mathematics') is just 'the orderly manipulations of symbols'. Or, as I prefer to phrase it, 'symbol games'.

That's

applied mathematics(or, perhaps,physics), anentirelydifferent beast with an entirely different epistemic status.Manipulations of symbols according to formal rules are the ontological basis, and our perception of "physical reality" results merely from our status as collections of symbols in the abstract Platonic realm that defines the convergent results of those manipulations, "existence" being merely how the algorithm feels from inside.

Yup, still makes sense to me!

I understand "symbols" to be a cognitive shorthand for our brains representation of structures in reality. I don't understand the meaning of the word "symbols" in the abstract, without a brain to interpret them with and map them onto reality.

This doesn't really explain anything to me, it just sounds like wisdom.

Think in terms of LISP gensyms - objects which themselves support only one operation, ==. The only thing we can say about (rg45t) is that it's the same as (rg45t) but not the same as (2qox), whereas we think we know what (forall) means (in the game of set theory) - in fact the only reason (forall) has a meaning is because some of our symbol-manipulating rules mention it.

As I understand it ec429’s intuition goes a bit like this:

Take P1, a program that serially computes the digits in the decimal expansion of π. Even if it’s the first time in the history of the universe that that program is run, it doesn’t feel like the person who ran the program (or the computer itself) created that sequence of digits. It feels like that sequence “always existed” (in fact, it feels like it “exists” regardless of running the program, or the existence of the Universe and the time flow it contains), and running the program just led to discovering its precise shape.(#)

Now take P2, a program that computes (deterministically) a simulation of, say, a human observer in a universe locally similar(##) to ours, but perhaps slightly different( ###) to remove indexing uncertainty. Applying intuition directly to P2, it feels that the simulation isn’t a real world, and whatever the observer inside feels and thinks (including about “existence”) is kind of “fake”; i.e., it feels like we’re creating it, and it wouldn’t exist if we didn’t run the program.

But there is actually no obvious difference from P1: the exact results of what happens inside P2, including the feelings and thoughts of the observer, are predetermined, and are exclusively the consequence of a series of symbolic manipulations or “equation solving” of the exact same kind as those that “generate” the decimals of π.

So either: 1) we are “creating” the sequence of decimals of π whenever we (first? or every time?) compute it, and if so we would also “create” the simulated world when we run P2, or 2) the sequence of digits in the expansion of π “exists” indifferently of us (and even our universe), and we merely discover (or embody) it when we compute it, and if so the simulated world of P2 also “exists” indifferently of us, and we simply discover (or embody) it when we execute P2.

I think ec429 “sides” with the first intuition, and you tend more towards the second. I just noticed I am confused.

(I kind of give a bit more weight to the first intuition, since P2 has a lot more going on to confuse my intuitions. But still, there’s no obvious reason why intuitions of my brain about abstract things like the existence of a particular sequence of numbers might match anything “real”.)

(#: This intuition is not necessarily universal, it’s just what I think is at the source ec429’s post.)

(##: For example, a completely deterministic program that uses 10^5 bit numbers to simulate all particles in a kilometer-wide radius copy of our world around, say, you at some point while reading this post, with a ridiculously high-quality pseudo-random number generator used to select a single Everett “slice”, and with a simple boundary chosen such that conditions inside the bubble remain livable for a few hours. This (or something very like it, I didn’t think too long about the exponents) is probably implementable with Jupiter-brain-class technology in our universe even with non-augumented-human–written software, not necessarily in “real-time”, and it’s hard to argue that the observer wouldn’t be really a human, at least while the simulation is running.)

(###: E.g., a red cat walks teleports inside the bubble when it didn’t in the “real” world. For extra fun, imagine that the simulated human thinks about what it means to exist while this happens.)

No, I'd say nearer the second - the

mathematical expressionof the world of P2 "exists" indifferently of us, and has just as much "existence" as we do. Rocks and trees and leptons, and their equivalents in P2-world, however, don't "exist"; only their corresponding 'pieces of math' flowing through the equations can be said to "exist".I don’t quite get what you mean, then. If the various “pieces of math” describe no more and no less than exactly the rocks and trees and leptons, how can one distinguish between the two?

Would you say the math of “x^2 + y^2 = r^2” exists but circles don’t?

Indeed. Circles are merely a map-tool geometers use to understand the underlying territory of Euclidean geometry, which is precisely real vector spaces (which can be studied axiomatically without ever using the word 'circle'). So, circles don't exist, but {x \in R² : |x|=r} does. (Plane geometry is one

modelof the formal system)And how exactly would you define the word “circle” other than {X \in R² : |x|=r}?

(In other words, if a geometric locus of points in a plane equidistant to a certain point exists, but circles don’t, the two are different; what is then the latter?)

The locus exists, as a mathematical object (it's the

string"{x \in R²: |x|=r}", not theset{x \in R² : |x|=r}). The "circle" on the other hand is a collection of points. You can apply syntactic (ie. mathematical) operators to a mathematical object; you can't apply syntactic operators to a collection of points. It is syntactic systems and their productions (ie. mathematical systems and their strings) which exist.Hmm. I’m not quite sure I understand why

abstractsymbols, strings and manipulations of those must exist in the a sense in whichabstractpoints, sets of points and manipulations of those don’t, nor am I quite sure why exactly one can’t do “syntactic” operations with points and sets rather than symbols.In my mind cellular automatons look very much like “syntactic manipulation of strings of symbols” right now, and I can’t quite tell why points etc. shouldn’t look the same, other than being continuous. And I’m pretty sure there’s someone out there doing (meta-)math using languages with variously infinite numbers of symbols arranged in variously infinite strings and manipulated by variously infinite syntactic rule sets applied a variously infinite number of times... In fact, rather than being convenient for different applications, I can’t quite tell what existence-relevant differences there are between those. Or in what way rule-based manipulations strings of symbols are “syntactic” and rule-based manipulations of sets of points aren’t—except for the fact that one is easy to implement by humans. In other words, how is compass and straightedge construction not syntactical?

(In terms of the tree-falling-in-the-forest problem, I’m not arguing about what sounds are, I’m just listing why I don’t understand what you mean by sound, in our case “existence”.)

[ETA. By “variously infinite” above I meant “infinite, with various cardinalities”. For the benefit of any future readers, note that I don’t know much about those other than very basic distinctions between countable and uncountable.]

Oh, I'm willing to admit variously infinite numbers of applications of the rules... that's why transfinite induction doesn't bother me in the slightest.

But, my objection to the existence of abstract points is: what's the definition of a point? It's defined by

what it does, by duck-typing. For instance, a point in R² is an ordered pair of reals. Now, you could say "an ordered pair (x,y) is the set {x,{x,y}}", but that's silly, that's not what an ordered pairis, it's just a construction that exhibits the required behaviour: namely, a constructor from two input values, and an equality axiom "(a,b)==(c,d) iff a==c and b==d". Yet, from a formal perspective at least, there are manymodelsof those axiomata, and it's absurd to claim that any one of those is what a point "is" - far more sensible to say that the point "is" its axiomata. Since those axiomata essentially consist of a list of valid string-rewriting rules (like (a,b)==(c,d) |- a==c), they are directly and explicitly syntactic.Perhaps, indeed, there is a system more fundamental to mathematics than syntactics - but given that the classes of formal languages even over finite strings are "variously infinite" (since language classes are equivalent to computability classes, by something Curry-Howardesque), it seems to me that, by accepting variously infinite strings and running-times, one should find that all mathematical systems are inherently syntactic in nature.

Sadly this is difficult to prove, as all our existing formal methods are themselves explicitly syntactic and thus anything we can express formally by current means, we can express as syntax. If materialistic and mechanistic ideas about the nature of consciousness are valid, then in fact any mathematics conceivable by human thought are susceptible to syntactic interpretation (for, ultimately, there exists a "validity" predicate over mathematical deductions, and assuming that validity predicate is constant in all factors other than the mathematical deduction itself (which assumption I believe to hold, as I am a Platonist), that predicate has a syntactic expression though possibly one derived via the physics of the brain). This does not, however, rule out the possibility that there are things we might want to call 'formal systems' which are not syntactic in nature. It is my belief - and nothing more than that - that such things do not exist.

These might be stupid questions, but I’m encouraged by a recent post to ask them:

Doesn’t that apply to syntactic methods, too? It was my understanding that the symbols, strings and transformation rules don’t quite have a definition except for duck typing, i.e. “symbols are things that can be recognized as identical or distinct from each other”. (In fact, in at least one of the courses I took the teacher explicitly said something like “symbols are not defined”, though I don’t know if that is “common terminology” or just him or her being not quite sure how to explain their “abstractness”.)

And the phrase about ordered pairs applies just as well to ordered strings in syntax, doesn’t it? Isn’t the most common model of “strings” the Lisp-like pair-of-symbol-and-pair-of-symbol-and...?

Oh, wait a minute. Perhaps I got it. Is the following a fair summary of your attitude?

We can only reason rigorously by syntactic methods (at least, it’s the best we have). To reason about the “real world” we must model it syntactically, use syntactic methods for reasoning (produce allowed derivations), then “translate back” the conclusions to “real world” terms. The modelling part can be done in many ways—we can translate the properties of what we model in many ways—but a certain syntactic system has a unique non-ambiguous set of derivations, therefore the things we model from the “real world” are not quite real, only the syntax is.I think that's a very good summary indeed, in particular that the "

unique non-ambiguous set of derivations" is what imbues the syntax with 'reality'.Symbols are indeed not defined, but the only means we have of duck-typing symbols is to do so symbolically (a symbol S is an object supporting an equality operator = with other symbols). You mention Lisp; the best mental model of symbols is Lisp gensyms (which, again, are objects supporting only one operator, equality).

conses of conses are indeed a common model of strings, but I'm not sure whether that matters - we're interested in the syntax itself considered abstractly, rather than any representation of the syntax. Since ad-hoc infinite regress is not allowed, we must take something as primal (just as formal mathematics takes the 'set' as primal and constructs everything from set theory) and that is what I do with syntax.

As mathematics starts with axioms about sets and inference rules about sets, so I begin with meta-axioms about syntax and meta-inference rules about syntax. (I then - somewhat reflexively - consider meta²-axioms, then transfinitely induct. It's a habit I've developed lately; a current project of mine is to work out how large a large ordinal kappa must be such that meta^kappa -syntax will prove the existence of ordinals larger than kappa, and then (by transfinite recursion shorter than kappa) prove the existence of [a given large cardinal, or the Von Neumann universe, or some other desired 'big' entity]. But that's a topic for another post, I fear)

What difference, if any, is there between adopting your definition of reality/existence, and banning the words reality/existence etc and using some other word for the concept you defined, other than miscommunication with people who use another concept of existence?

I think this is relevant to some of the discussion here: David Chalmers, The Matrix as Metaphysics.

Chalmers argues that discovering that the simulation hypothesis was true would not show our beliefs about the existence of physical objects to be

false, but merely indicate a surprising truth about the nature of reality. So he would say that yes, the rockdoesexist even if the simulation hypothesis is true.I think I have thought of, not a disproof, but evidence against what you're suggesting.

For every regular mathematical system, there exist infinitely many slight non-regular alterations of the system, implying that, if you are correct, there are infinitely more non-regular universes than regular universes. Under the principle that there's nothing special about our location in space/time/etc, we ought to find ourselves living in a "typical" universe, that is, a non-regular one. Yet, our universe appears to be highly regular.

Maybe you can argue that most such universes have very

smallirregularities...Suppose there are some cosmic rays, and A gets a bit wrong here and there, randomly. A is no longer simulating B, A is simulating B'. Further, assume that chaotic effects ensure that B' soon radically diverges from B. Do you claim that B' has an existence separate from the one A is giving it, that the "causal relationship" goes from B' -> A? Are you claiming that all possible mistaken simulations of all possible sets of physical laws somehow exist platonically?

It's not clear to me what this system you're describing gains you. What predictions does it cause you to make differently? (i.e., how does it pay rent?)

Inasmuch as a mistaken simulation of B is a simulation of "B plus the arbitrary law that 'foo happens at time t'", then yes, B' exists platonically. Consider B = "x'' + x = 0, x(0)=x'(0)=0", B' = "x'' + x = δ(x-1), x(0)=x'(0)=0". These are both coherent mathematical models; they model different things, but B' still exists platonically. A cosmic ray could indeed make an ostensible model of B model B'.

It's observationally equivalent to the hypothesis of manifest reality, but strictly simpler (it removes unnecessary ontological elements, and converts uncertainty about physical law into indexical uncertainty). Like Many Worlds, really.

Does your model differ from "Everything that could exist, does"? I can imagine setting up a simulator for a universe just like ours, except that every time a planet coalesces, it's spontaneously replaced with a kangaroo. Does the fact that I can describe such a system imply (to you) that it actually exists? That without me even actually making a simulation of it, multitudes of kangaroos are each experiencing a few brief seconds of asphyxiation around their stars?

Well, it's making me think, anyway.

Not quite. Every mathematical structure (strictly, every syntactic language) that could exist, does (and causes its corresponding apparent physical reality to appear to exist) - so the kangaroos do indeed exist (and did so even before you thought of them). After all, how could you have an effect on the kangaroos merely by simulating them? (I maintain that the causal relation runs from the kangaroos to you, not the other way round).

It may well be impossible to devise an example of something that doesn't exist (because, if we can think about it, it is modellable), but that doesn't mean to say that things that don't exist, um, don't exist... I mean, {X : not exist(X)} need not be empty, it's just that we can't exhibit any of its elements. It might be provably empty, but I have no proof here and can't imagine what form one would take.

Also, you have to be a bit careful about the causal relations; if you start thinking of them as one formal system modelling another, you get into Gödel-space. Hence the Syntacticism post; I think it's accurate to say that one formal system models a

quotationof another, but I'm not sure of that.Existence is a bit odd.

We start off by saying that the universe that I can sense exists, along with me. I carry on to allow existence to those bits of the universe that I can't directly sense, but it seems reasonable to infer are present from the physics of the universe.

So existence seems straightforward. But what about non-existence? Particularly in the context of a possible multiverse?

I think what happens is that people figuratively draw out in their minds the whole class IV multiverse that Tegmark talked about, and ask themselves how they would test for the non-existence of a part of it. The answer is that you can't - experiments can only manipulate and measure existing things using other existing things. You never get to experiment on a non-existent thing.

This means that the whole question "What exists?" is not directly a scientific one at all. Well, partially - you can do science on some of the things that do exist, but you don't ever get to do science on any of the things that don't exist. All you can ever confirm is that you can't see them from where you stand.

I think the right approach is to drop the 'exists' word altogether in this context, and instead ask yourself how to describe the universe using the usual approach of looking for the theory which is simplest.

I can see no justification whatever for that method. What's so special about you, that you imbue things with "existing-ness" by sensing them? Surely that's an egregious Mind Projection Fallacy? "It's in my map, so it must be in some territory somewhere"?

I wasn't actually justifying the move - merely saying this is how the concept of existence comes about. My final sentence sums up what I actually think - that 'exists' is a deeply misleading concept in situations such as this, and should be confined to folk philosophy, where it's commonly used meaning is sufficient.

Getting rid of the 'exists' concept is more or less what I'm trying to

do- or rather, show that if you have an 'exists' concept such that ¬exists(infinite sets) then your 'exists' concept is incoherent; moreover, that an 'exists' defined by exists(our Universe) and ¬exists(everything else) is not an important concept and should be detached from the connotations the savannah brain likes to associate with things that 'exist'."exist" doesn't have a referent. Any attempt to define it will either be special pleading (my universe is

special, it"exists", because it'sthe one I live in!), or will give a definition that applies equally to all mathematical structures.Surely can't be exactly what you mean, as exists(our Univese) and ¬exists(everything else) seems coherent if rather unlikely, and seems consistent at our present state of knowledge with ¬exists(infinite sets).

It seems your 'exists' concept is pretty much indistinguishable from 'logically coherent', and that that's the whole point you're trying to make - that we're in no position to distinguish these, and should simply abandon the 'exists'.

I would dispute this, on the grounds that my deductions in formal systems come from somewhere that has a causal relation to my brain - the formal system causes me to be more likely to deduce the things which are valid deductions than the things that aren't. So, if I 'exist', I maintain that the formal systems have to 'exist' too, unless you're happy with 'existing' things being causally influenced by 'non-existing' things - in which case there's not a lot of point in asserting that ¬exists(infinite sets). A definition of 'exists' which doesn't satisfy my coherence requirements is, I am attempting to argue, simply a means of sneaking in connotations.

A more conservative approach is to point out that some of our (mathematical) models are useful: they let us predict what happens when we perform an experiment, while others are not useful in that sense (though they can still can help you get through a grad school, write a grant application, gather a few karma points when posted here, or maybe even found a new church, if your model is untestable but convincing enough).

If you are a natural scientist, it's the "useful" models that should matter to you, while if you are a philosopher, there is little distinction. Your musings are clearly of the latter kind.

I would need to know more about physics to even understand how this differs from many-worlds.

Tegmark IV and many worlds have little to nothing to do with each other.

alternateset of equations is equally "real" as our own, that a concept existing in abstract mathematical form and a physical structure existing in reality is one and the same thing. Effectively it attempts to explain reality by arguing the word reality has no meaning.Sorry for this inane comment.

If people make random nonsensical comments and then immediately retract them without explanation just so they can troll and spam but at the same time avoid downvoting, then I have to go downvote two other comments they made, in punishment both for the nonsensical comment and for the abuse of the "retract' action.

I didn't realize others could see my retracted comments. I guess I don't know what the purpose is of 'retract'.

Having made a random, nonsensical comment what is the best action to take to mitigate the damage of it?

After a comment is retracted, it can be deleted. The "delete" button appears after you reload the page with a retracted comment. "Retraction", by blocking the downvoting, exists to incentivise retaining of the original text where there's already discussion that references it.

"Retract" makes it clear that you have changed your view about your comment, but leaves it available for others to read. If you wish to remove the text as well you can just press edit and delete it, possibly replacing it with an apologatory message.