No , because a Tegmark level IV mathematical Universe—where the apparent physical universe is just a small.part of the mathematical one—isn’t obviously contradictory. (It might be better to say physical objects aren’t inherently non mathematical).
so not sure how that would work. But let’s say it’s something we can think up. It’s just not true of our current situation in the universe. I see you chose it as a counter balance to Wigner’s point about unreasonable effectiveness insofar that ‘why expect mathematics to work at all? It’s the way it is, so proposing a less effective situation isn’t reasonable either’. I hear you, but I think you are not following—I didn’t set out to resolve a mystery about a preconceived notion of what’s reasonable (based on a biased thought), ala Wigner, I set out to explore the mechanism of Ansatz and how it could work in this kind of formalism.
To argue for realism against anti realism, you need to show that “ansatz” can’t work under anti realism, which you haven’t done. If the physical universe is at least partly describable mathematically, , then random guessing at mathematical models will work occssionally. So you dont need to assume more for Anstatz than for UEM.
I thought it was neat that a possible resolution to the ‘unreasonableness’ falls out for free, and so, pointed it out (in this formalism, at least). So I am arguing that it is very reasonable, and could be a helpful way of resolving a possible preconceived aghastness ala > thought it was neat that a possible resolution to the ‘unreasonableness’ falls out for free,
It doesn’t because, as I explained, the existence of a mathematical.universe implies nothing about the mathematical describability of a separate physical universe. (A Tegmark solution does, but you have rejected it!)
and so, pointed it out (in this formalism, at least). So I am arguing that it is very reasonable, and could be a helpful way of resolving a possible preconceived aghastness ala Wigner.
You didn’t follow me—I didn’t claim mathematical realism explains it, I assumed mathematical realism (I put some sentences in there describing why I think its good) but from that point, assuming mathematical realism, I develop a formalism, and that formalism has this neat result.
Which result?.If it’s not a solution to UEM, why bring it up?
I make this point in the OP, if you read. In fact, we just hold it as a belief that the universe is inherently (or maybe only mostly) reasonable and structured. It’s worked quite well so far. But whoMs to stop it from being, rock bottom, having even a single component thatMa completely immune to it? I venture to suggest there are some aspects of even non-rel QM that suggest as such, and once of my friends Cael who did his PhD in mathematical physics got quite deep in this area, and was starting to feel that the whole idea of there being objects that ‘have attributes’ might subtly break down at some point.
Well, it might, but I don’t quite.see what that has to everything else.
Anyway, I’m not sure what you are pointing it
Im trying to find out why you even mentioned UEM. Solips ism?
“That isn’t a gotcha against mathematical anti-realists, though, because they are well aware of it. Why did you bring it up?”
Well, I’m interested to see how it is explained in a totally naturalistic way from others, hence why I bring it up. The main explanation I hear is that mathematics is a kind of a ‘special fiction’ that is shared in the minds of people/humans and doesn’t have a real existence per se. I’m not sure how this goes without leading to a solipsism.
I don’t see how it leads to solipsism. “So we actually get a clear solipsism if we go too far with that approach” doesn’t explain it either. It isn’t clea r.
“If there is a material universe as well, obviously maths is outside it.”
!!!! That’s not at all obvious, in fact, I’d never heard it before until I began looking into this. If it’s obvious now to many people, that’s great! Perhaps the cultural mindset is different compared to my younger days. Certainly the most common view I run into among academics ‘spoken’ is that maths is included in the physical universe as a kind of ‘convenient fiction’
and the physical universe is there is, but then in reality they often kind of tacitly adopted this kind of mathematical realism—I wanted to explore why there’s some unwillingness to face this hypocrisy Of
You seem to be blurring “whether mathematical realism is true” and “what are they implications of MR”. If MR is true , then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical.
so it got me thinking down this track. Anyway, the punch of my article is really some fancy machinery that shows that if there is a material universe then maths is outside it,
Assuming realism...?
“But what does that have to do with the supernatural (particularly in the Carrier sense)? Plenty of naturalists believe in mathematical realism”
I’m not sure I know what a Carrier sense for supernatural is, but what this has to do with the supernatural
The first google match for “Carrier supernatural” is
is purely the literal definition of the universe and if something can be outside of it, supervening (ie structure and equations that sit outside it).
Well, maybe aren’t using, or don’t care about the literal definition.
If plenty of naturalists believe in mathematical realism, as I have argued, there’s an inherent contradiction that’s always unspoken
Between mathematical realism and mathematical fictionalism , or between mathematical realism and naturalism?
Re: the latter.. If they think of the supernatural as gods and ghosts, as most people to, then there isn’t because mathematical realism doesn’t entail anything like that. I think the ghosts and ghoulies definition is what people care about.
“No , because a Tegmark level IV mathematical Universe—where the apparent physical universe is just a small.part of the mathematical one—isn’t obviously contradictory. (It might be better to say physical objects aren’t inherently non mathematical).”
For a mathematical universe, it only isn’t obviously contradictory if it is constrained in a way to avoid the Cantor problem. But what I mean is, people are clearly accessing many mathematical truths beyond that that don’t specifically exhibit in the universe,
(aside: unless you count the thought process itself as the exhibition, but then as its a question of will and enterprise, it could be arbitrarily expanded with effort which still puts the bound at a level where we could say the universe is expanding into something, the thing which it is expanding into suffers the Cantor problem.)
And a point here not mentioned is your comment is that these mathematical objects if coupled to a normal universe still exhibit different properties: they don’t have a location, they are timeless, they don’t exhibit many of the usual attributes of physical objects etc.
The statement of physical objects not inherently being non-mathematical is a good one—I like that. Clearly there is a deep link between mathematics and nature as the mathematical attributes get exhibited in many places.
“To argue for realism against anti realism, you need to show that “ansatz” can’t work under anti realism, which you haven’t done. If the physical universe is at least partly describable mathematically, , then random guessing at mathematical models will work occssionally. So you dont need to assume more for Anstatz than for UEM”
I really don’t need to show that—as mentioned previously, I am not attempting to demonstrate Mathematical Realism, but I give some background as to why I chose it, and then I assume it to explore some consequences. Assuming something different would be interesting but its a different project and beyond the scope of this work at present. I haven’t thought of how to construct an Ansatz in a framework built from an arbitrary metaphysic—perhaps you can’t, perhaps each one is done on a case by case basis. I simply chose a metaphysic that seemed the most sensible and aligned most closely of the working ethos of myself and colleagues, working in physics—it’s a starting point.
I would anticipate that you could still construct an Ansatz approach in a different framework—and that there are some caveats that need to be resolved ie it couldnt be constructed in a naive way—it certainly doesn’t strike me that the Ansatz part of all this is the thing that would threaten non-realism necessarily but I haven’t explored that.
And I agree re your comment randomly guessing models. I certainly don’t know for certain, nor prove, that there is not a component of the universe not modellable in this way. Before I started this project, me and a colleague often thought about what if it is not.
(Cont’d) apologies, this is taking some time and I will do this in parts as I will run out of time here and there. Bear with me.
“It doesn’t because, as I explained, the existence of a mathematical.universe implies nothing about the mathematical describability of a separate physical universe. (A Tegmark solution does, but you have rejected it!)”
Did you read the paper? It’s not the existence of a mathematical universe that is used to show it, but, given the framework, I use a cardinality argument—so there’s more work and proofs and theorems in the paper—I just summarise the cliff notes in this post for the lay audience. What I do is use a very general statistical observation only, rather than trying to link up individual objects to theorems, I look at the relative size of the spaces. What I have explored is the connection between a separate universe in a framework where the mathematical principles can be used to describe things in many universes, the reverse-Epiphenomenal view.
Also, I haven’t totally rejected a Tegmark view, it just needs to be subtly qualified—ie the mathematical parts tied together with the universe can’t be unscoped, one is limited by Cantor’s Theorem here.
“Which result?.If it’s not a solution to UEM, why bring it up?”
The result: Equation (54) in the paper. I brought it up because it was interesting, and I hadn’t seen a cardinality argument before.
If by UEM you mean Universal Existing Mathematics, well that’s not what the work is trying to demonstrate. It seems like you intended this work to be about proving UEM, and are frustrated that it doesn’t. I’m confused because that’s not what I set out to do and it’s interesting, but not the topic I am looking at.
“Well, it might, but I don’t quite.see what that has to everything else.”
The reason for my aside, was it was an example of what could be true—I’m trying to say I don’t necessarily hold that the universe is always structured and reasonable, it might not be. What I did was assume it, and show the result. I certainly don’t prove it. It would be interesting to get more data and explore these things. But in the case where the universe is reasonable all the way down, then this would hold. It seems like you wanted me to prove or disprove something which isn’t the topic I set out to do. You can do it if you want. (Cite my paper though :)
“Im trying to find out why you even mentioned UEM. Solips ism?”
I’m really not following you now. I adopt Mathematical Realism for the reasons stated, its a philosophy that’s reasonable and also aligns with the working ethos of physics and mathematics. What other popular flavor should I have chosen?
“I don’t see how it leads to solipsism. “So we actually get a clear solipsism if we go too far with that approach” doesn’t explain it either. It isn’t clea r.”
Ah ok, so the solipsism goes as follows. Note that this does not constitute a proof, it’s really more pointing out a metaphysical convenience, where physicalism on the other hand needs to jump through some more subtle hoops and goes a little against Occam’s Razor.
That is: in the specific version of physicalism that seems mathematical entities as nonreal, and fictions that exist in the human mind, then structures that pre-date humanity, like the mathematical groups associated with particle behavior in the early universe, couldn’t’ve existed back then. So I intuit that the actual structure is timeless and not incorporated easily into the universe that way. It’s not the only way, there are other versions of physicalism.
8. “You seem to be blurring “whether mathematical realism is true” and “what are they implications of MR”. If MR is true , then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical.”
I wasn’t able to perceive the blur between these two items in my quote: ”!!!! That’s not at all obvious, in fact, I’d never heard it before until I began looking into this. If it’s obvious now to many people, that’s great! Perhaps the cultural mindset is different compared to my younger days. Certainly the most common view I run into among academics ‘spoken’ is that maths is included in the physical universe as a kind of ‘convenient fiction’ and the physical universe is there is, but then in reality they often kind of tacitly adopted this kind of mathematical realism—I wanted to explore why there’s some unwillingness to face this hypocrisy… (etc)”
I’m simply relating to you what my experience has been in regard to the conversations I have had. The observation that some people will state they adopt a view, and then operate with another view, is not a commentary on whether MR is true or not, or whether I believe it to be true. It’s something that I noticed, and I have in this work taken a stance of MR being true.
I’ve tried to be as clear as I can, and I havent deviated from the same point that I, A) adopt MR (I give some context as to why I think it’s reasonable and a natural view, for my own part), and then B) I develop a formalism assuming it, in the format that I describe (there may be other variants) and then I work through some of the implications of the formalism. I feel like I’m repeating myself over and over to you, I’m not quite sure why it’s not clear.
When you state ‘If MR is true, then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical’ doesn’t follow, in my view. Why would MR being true mean that the mathematical universe is necessarily bigger than the physical universe? The only way I can see it being obvious is if you are defining the ‘mathematical universe’ as the physical universe and the mathematical part altogether. Perhaps that’s what you mean, but you didn’t state it.
In the work I have put together, you can see that what I am showing from the formalism is that extending the physical universe to encompass the mathematical truths runs into some practical issues, and instead defining the universe as being smaller than this, necessarily leaves out some objects that are now not in the universe.
9. “Assuming realism...?” As mentioned, yes the formalism takes MR as a backing metaphysic as previously stated.
Ah, I haven’t read this author before and didn’t realise Carrier was the name of an author, I had assumed it was a jargon coined on here perhaps. But having a brief look, I’m honestly not sure how to match my work to this definition of supernatural. Do I have to? I am anticipating my work can be standalone and not to try to use another person’s definition of supernatural. Here, I use it purely as ‘not in the universe’ where I’ve defined the universe in the way I describe in the work.
I mean, it’s interesting, but one thing that makes it difficult to do matchup with another work is some of the terms don’t appear to be carefully defined enough, e.g. “Tautologically a natural world is a world with nothing supernatural in it, and a supernatural world is a world with at least one supernatural thing in it.” It’s not clear to me what a ‘world’ is, here, or how it is intended to relate to the kinds of realities I am talking about.
11. “Well, maybe aren’t using, or don’t care about the literal definition.”
In that case, umm. I got nothing. If people are using different definitions or don’t care about the literal definition, it doesn’t really impact the meaning of the work I am trying to do here.
12. “Between mathematical realism and mathematical fictionalism , or between mathematical realism and naturalism?
The inherent contradiction I was meaning here was more the former: a Naturalist is bound into believing that the natural laws, mathematical principles governing nature (and so forth) are part of nature or an emergent reality, in some versions, it is present in our mind, and yet with the other hand, will operate as though there was a ‘math land’ where only some items from that apply to our universe. Both valid points of view but incompatible.
In the latter case, you could totally have a Naturalism that extends the physical part to encompass the abstract and mathematical part, as described above, though it need to be done carefully and some methods of doing that can result in contradictions (the ‘draw a box around everything’ scenario). I apologise for the confusion that these are two separate points.
13. “Re: the latter.. If they think of the supernatural as gods and ghosts, as most people to, then there isn’t because mathematical realism doesn’t entail anything like that. I think the ghosts and ghoulies definition is what people care about.”
I hear you, but then that mixes in some folklore aspects adding another dimension of complexity. In this post (ie not the article, but the post) what I noted was that this opens the door—once you admit one supernatural supervening, it demonstrates at least one case where it can occur. I’m of the view that the folklorish aspects in humanity don’t come from nothing, and while many of the folklore stories may not be true, they keep coming up in every culture. I’ve been thinking for a while that the difference between a ‘demon’ and a ‘mindset’ seems slight, and there might be some truth to the idea that the abstract has a more ‘real’ aspect and dimension than people are in the habit of believing right now. That the mind is participating in real, genuine discovery and creation when it deals with mathematics.
And also, it doesn’t matter what people think or care about, let’s work out the truth first, and then we need to believe it, regardless of how uncomfortable it is, or what previous propaganda says, or it has a bad reminding taste of some folklore. So many weird physics things that seem unbelievable and seem crazy I have had to accept over the years as truth. If it’s true it’s true, and I think these aspects are also talked about in the neurology book The Master and His Emissary by Iain McGilchrist and his follow up book ‘The Matter with Things’, which argues that both the ‘Reality-Out-There’ and the ‘Made-Up-Miraculously-By-Our-Minds’ views of reality are both false, and that there is a contribution from both the observer and the environment at the same time in creating reality. The reductionist view of nature to boil it down to something more objective, you can see, is actually highly stylized, attitude driven and not objective at all.
When you state ‘If MR is true, then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical’ doesn’t follow, in my view. Why would MR being true mean that the mathematical universe is necessarily bigger than the physical universe?
Because most maths isn’t physically applicable, as I stated, and you agreed.
. But having a brief look, I’m honestly not sure how to match my work to this definition of supernatural.
You have a communication issue, because you are not using “supernatural” in the expected way, and a PR issue, because a lot of your intended audience are going to reject the supernatural out of hand. Whence the downvoting.
Do I have to? I am anticipating my work can be standalone and not to try to use another person’s definition of supernatural. Here, I use it purely as ‘not in the universe’ where I’ve defined the universe in the way I describe in the work.
You need to communicate clearly , and you don’t need to repell the reader
“Between mathematical realism and mathematical fictionalism , or between mathematical realism and naturalism?
The inherent contradiction I was meaning here was more the former: a Naturalist is bound into believing that the natural laws, mathematical principles governing nature (and so forth) are part of nature or an emergent reality,
Again, that’s not the same thing. The existence of X-ishly describable entities doesn’t imply the existence of free-standing X’s. For instance, we can describe the colours of external objects using the trichromic RGB system , but it’s definitely not out there.
Context: When you state ‘If MR is true, then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical’ doesn’t follow, in my view. Why would MR being true mean that the mathematical universe is necessarily bigger than the physical universe?
Comment 1: “Because most maths isn’t physically applicable, as I stated, and you agreed.”
Response 1: I do agree that most maths isn’t physically applicable, but that doesn’t mean that for MR, the MU is obviously bigger (to clarify, for MU here, do you mean physical+maths, I am assuming not). For example, I might have many physical objects in my universe, and not all being mapped to by a mathematical abstraction. I have no way of ensuring that the universe is all totally mapped to. I make a supposition that in physics, we hold a view that it can (or should be). But I don’t know for sure, and so the relative sizes of physical + mathematical parts is hard to define. It may be the case the the mathematical part is indeed larger, but the fact that most maths isn’t physical doesn’t guarantee it, it would be something to do with limits on the size of the physical universe, and/or the scope of mathematics obtruding into it. Maybe most physical doesn’t get mapped to (though, I don’t believe that currently, it could definitely be proposed).
. But having a brief look, I’m honestly not sure how to match my work to this definition of supernatural.
Comment 2: “You have a communication issue, because you are not using “supernatural” in the expected way, and a PR issue, because a lot of your intended audience are going to reject the supernatural out of hand. Whence the downvoting.”
Response 2: Thank you for the view. The way I see it though, it is actually a good communication method, in that I have excited some commentary and engagement from the community—such as yourself—you have been very generous with your engagement. The rejection out-of-hand though accidentally demonstrates that the audience might not have been as attentive as they might pride themselves on, however, which itself is a useful insight to note.
Do I have to? I am anticipating my work can be standalone and not to try to use another person’s definition of supernatural. Here, I use it purely as ‘not in the universe’ where I’ve defined the universe in the way I describe in the work.
Comment 3: “You need to communicate clearly , and you don’t need to repell the reader”
Response 3: I am attempting to communicate as best I can, and am limited of course by my competence. Apologies if it doesn’t come up to scratch—I am doing my best. I also am not intending to repel the reader, but get some engagement, which was successful.
Also, the ‘do I have to’ was in the context of whether i need to match my work to this definition of supernatural, not based on communicating clearly, per se. I wasn’t aware of the work, but how else do I generate discussion to get some improvement from the lesswrong community? I have to start somewhere. I was as clear as my faculties allow. I tried to define the supernatural the way I see it. A comparison of that view and another work seems like a different topic beyond the scope of this post.
“Between mathematical realism and mathematical fictionalism , or between mathematical realism and naturalism?
The inherent contradiction I was meaning here was more the former: a Naturalist is bound into believing that the natural laws, mathematical principles governing nature (and so forth) are part of nature or an emergent reality,
Comment 4: Again, that’s not the same thing. The existence of X-ishly describable entities doesn’t imply the existence of free-standing X’s. For instance, we can describe the colours of external objects using the trichromic RGB system , but it’s definitely not out there.
Response 4: I didn’t say that the existence of X-describable entities implies free standing X’s. The idea of free standing X’s, ie a kind of Platonism is not something i set about to prove. I believe I assumed it as a starting point, and wanted to see how far I could get with it, as an exercise. So I wouldn’t argue it that way. I do state in the above quote that the Naturalist was bound to believing natural laws (by definition) and that I take it to mean that mathematical principles governing nature emerge from this same (physical) universe, as opposed to a more Platonic view of Mathematical Realism (ie. “out there” as you put it). Is that untrue? In regard to such a Platonic view, I would take it that the colours of objects using an RGB system, which are both concepts (the colours, and the RGB system) are abstractions that have an existence, and would be included as one of the abstractions in my formalism, that then get projected down in a reverse-epiphenomenal way, onto the physical-world object. (ie they are attributes of it, and attributes are abstractions).
I haven’t thought of how to construct an Ansatz in a framework built from an arbitrary metaphysic
I have. There’s almost nothing to it, if Ansatz means nothing more than some guessed-at mathematical descriptions turning out to be right.. and that is the . description of Ansaztz you give here:-
1 Introduction
The study of physics inherently requires both scientific observation and philosophy. The ten-
ants of science, and its axioms of operation, are not themselves scientific statements, but philo-
sophical statements. Historically, the profound philosophical insight precipitating the birth of
physics was that scientific observations and philosophical constructs, such as logic and reason-
ing, could be married together in a way that allowed one to make predictions of observations (in
science) based on theorems and proofs (in logic—a branch of philosophy), rather than simply
to collect data on phenomena without interpretation. This natural philosophy requires a philo-
sophical ‘leap’, in which one makes an assumption (or guess) about what abstract framework
applies most correctly. Such a leap, called Ansatz, is usually arrived at through inspiration
and an integrated usage of faculties of the mind, rather than a programmatic application of
certain axioms. Nevertheless, once a set of fundamental principles are decided upon, a subse-
quent programmatic approach allows enumeration of the details of the ensuing formalism for
the purposes of such an application. It seems prudent to apply a programmatic approach to
the notion of Ansatz itself and to clarify its process metaphysically, in order to gain a deeper
understanding of how it is used in practice in science; but first of all, let us begin with the
inspiration.
2 A metaphysical approach
In this work, a programme is laid out for addressing the philosophical mechanism of Ansatz.
In physics in general, a scientific prediction is made firstly by arriving at a principle, usually
at least partly mathematical in nature. The mathematical formulation is then ‘guessed’ to
hold in particular physical situations. The key philosophical process involved is exactly this
‘projecting’ or ‘matching’ of the self-contained mathematical formulation with the supposed
underlying principles of the universe. No proof is deemed possible outside the mathematical
framework, since proof, as an abstract entity, is an inherent feature of a mathematical (and
philosophical) viewpoint. Indeed, it is difficult to imagine what tools a proof-like verification
in a non-mathematical context may use or require.
..then, so long as some things are mathematically describable, some mathematical descriptions will describe them, even if guessed randomly.
(We don’t know where Ansatze come from in a detailed way, but it’s hard to see why that would need a supernatural/metaphsyical explanation, since we don’t know where “think of a number comes from”, but don’;t doubt that it is an ordinary psychological process. The whole rhetoric surrounding Ansatz, or guessing as I like to call it, is overblown, IMO).
But Wigner brings in further issues—the issue that a guessed-at mathmaticlal structure which is intended that is intended to describe one phenomenon, can be applicable to others. And you mention , in relation to Dirac’s relativistc wave equation, the ability to make successful novel predictions..
The various sub-problems have various possible solutions.
An ontology where the universe is based on a set of small set of rules explains the unreasonable effectiveness well enough: since each rule has to cover a lot of ground, each rule has multiple applications. And such an ontology is already fairly standard.
There’s also an underlying problem that saying “I can solve X” has two meanings: “My assumptions are the only solutions to X” and “I have the latest in a long line of putative solutions” It is not enough to succeed, others must fail.
Context: I haven’t thought of how to construct an Ansatz in a framework built from an arbitrary metaphysic
Comment 1: “I have. There’s almost nothing to it, if Ansatz means nothing more than some guessed-at mathematical descriptions turning out to be right.. and that is the . description of Ansaztz you give here:- (etc) ..then, so long as some things are mathematically describable, some mathematical descriptions will describe them, even if guessed randomly”
Response 1: Hang on, what I mean is, constructing an Ansatz completely from scratch, without any assumed structure doesn’t sound like something there would be ‘nothing to it’ - I would expect that if you have one, you’d need to be careful not to accidentally smuggle in an assumed concept from the get-go, which hasn’t been demonstrated yet—it’s hard to come to any logical machinery or systems from scratch without assuming something, without any structure or rules or symbols at all. Even if it is very simple, you have to start from somewhere. I tried to keep mine very general, and a few items of structure were added as minimally as possible. But what I mean is, the very concept of an Ansatz itself is automatically couched in some framework—I don’t think one can have a concept unless one at least has a framework for the concept to be part of, or to exist in, so I would assume to even invoke the concept, a framework (even a skeleton one) has been assumed.
Comment 2: “(We don’t know where Ansatze come from in a detailed way, but it’s hard to see why that would need a supernatural/metaphsyical explanation, since we don’t know where “think of a number comes from”, but don’;t doubt that it is an ordinary psychological process. The whole rhetoric surrounding Ansatz, or guessing as I like to call it, is overblown, IMO).”
Response 2: In terms of where Ansatze come from, I don’t think we do know quite where it comes from but we don’t need to know for the purpose of this investigation yet—it’s simply enough that we require logic to exist, and for there to be an abstract concept that can be invoked—very little else in the proof was assumed. The supernatural explanation (again, being careful to define what I mean here by supernatural, that is being outside the physical universe) comes about naturally, with only some minimal rules of logic being invoked. We might not know ‘where’ “think of a number” comes from, but we do know that the number is consistently definable, it ‘exists’ (that’s taken based on an MR viewpoint though), and it gets instantiated a lot, in the physical universe—ie the physical objects obey (and have an intimate relationship with) these mathematical objects.
In terms of the psychological process by which we access it, the psychology would be developed from brain structures, and those are based on proteins, based on info from genes, on chemistry, on physics, down to the smallest particle, so at every level, we have seen a great deal of natural processes are respecting mathematics, and we can write down these laws. So it would come at no surprise that our brains are also structured and follow processes. But, you wouldn’t argue that if the brain was destroyed, that the concepts being referred to by some maths would be destroyed, nor would an atom being destroyed mean the concepts of mathematical groups an equations of motion would be destroyed. Surely those are just all instances but not the thing being referred to itself. (ie they are not the mathematical truths themselves, as those truths turn up in all sorts of places).
Apologies if the rhetoric seems overblown—can you specify in what way? As above, I haven’t quite got your view in mind re mathematical truths. It seems you can’t have no metaphysic, we all have a metaphysic in mind, just it might be undeclared or unexamined—so I am interested to learn yours—it seems yours, to you, seems preferable, but I am unclear of your statement of it.
Comment 3: “But Wigner brings in further issues—the issue that a guessed-at mathmaticlal structure which is intended that is intended to describe one phenomenon, can be applicable to others. And you mention , in relation to Dirac’s relativistc wave equation, the ability to make successful novel predictions.. The various sub-problems have various possible solutions. ”
Response 3: What are the issues raised by Wigner issues for? ie it seems consistent with the metaphysic I have adopted. Many different mathematical mechanisms can be applied to describe processes. It happens all the time in particle physics. There’s a concept of a ‘Representation’ of a group. Subatomic particles are arranged into Groups, which have a certain mathematical structure. But a group is quite general, and you can represent a group in different ways. One particular group might have many different representations, one using matrices, one using complex exponentials, all sorts. These representations have the group structure, but they add more, adding specificity, and are called vector spaces. You could use one machinery to look into a physical phenomenon, or you could use another. Both could apply. Hence why it is hard to have a prescription for how to ‘select just the mathematical truths’ applicable to the physical universe so as to bolt them on and get a consistent P+M Universe (hence why I don’t go down that route).
Comment 4: “An ontology where the universe is based on a set of small set of rules explains the unreasonable effectiveness well enough: since each rule has to cover a lot of ground, each rule has multiple applications. And such an ontology is already fairly standard.”
Response 4: The universe being ‘based’ on a set of rules is an interesting phrase, as it does seem similar to my version of MR—ie that there are rules, and those can be talked about in a meta way, regardless of physical universe, and then the physical universe can be talked about as following those rules. I also agree, since it is the view I was expounding, that this leads to an explanation of the unreasonable effectiveness, but the way I said it was different—I just counted the countably-infinite number of possible abstractions that could apply to a phenomena in a physical universe, and the seemingly ‘smaller’ (more restricted) countably-infinite number of abstractions applying to phenomena that also are extant in a universe, and found them to be of the same Cantor cardinality.
Comment 5: “There’s also an underlying problem that saying “I can solve X” has two meanings: “My assumptions are the only solutions to X” and “I have the latest in a long line of putative solutions” It is not enough to succeed, others must fail.”
Response 5: It is true that a phrase like ‘I can solve X’ can have an ambiguity. I take it to mean the latter though, in other words, taking some assumptions, and working through some steps, one can arrive at a valid solution—which in and of itself has merit, without stating whether other approaches can get to the same point. One might point out the neatness (less ‘epicycles’) or more (or less) in line with Occam’s Razor to evaluate a purported solution after it is given—but I wouldn’t say that means a solution hasn’t been given. Certainly in my work I don’t state it’s the only way, it’s much smaller than that, as a claim—just that this way seems to hold up, seems neat, a lot of things fall out of it ‘for free’ (ie it solves the problem with low entropy, without over-engineering extra unnecessaries) and it also seems to align with thinking drawn from multiple different fields of knowledge which is usually a detective’s ‘hint’ in the right direction, during an investigation.
No , because a Tegmark level IV mathematical Universe—where the apparent physical universe is just a small.part of the mathematical one—isn’t obviously contradictory. (It might be better to say physical objects aren’t inherently non mathematical).
To argue for realism against anti realism, you need to show that “ansatz” can’t work under anti realism, which you haven’t done. If the physical universe is at least partly describable mathematically, , then random guessing at mathematical models will work occssionally. So you dont need to assume more for Anstatz than for UEM.
It doesn’t because, as I explained, the existence of a mathematical.universe implies nothing about the mathematical describability of a separate physical universe. (A Tegmark solution does, but you have rejected it!)
Which result?.If it’s not a solution to UEM, why bring it up?
Well, it might, but I don’t quite.see what that has to everything else.
Im trying to find out why you even mentioned UEM. Solips ism?
“That isn’t a gotcha against mathematical anti-realists, though, because they are well aware of it. Why did you bring it up?”
I don’t see how it leads to solipsism. “So we actually get a clear solipsism if we go too far with that approach” doesn’t explain it either. It isn’t clea r.
“If there is a material universe as well, obviously maths is outside it.”
You seem to be blurring “whether mathematical realism is true” and “what are they implications of MR”. If MR is true , then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical.
Assuming realism...?
The first google match for “Carrier supernatural” is
[https://www.richardcarrier.info/archives/7340(https://www.richardcarrier.info/archives/7340)
Well, maybe aren’t using, or don’t care about the literal definition.
Between mathematical realism and mathematical fictionalism , or between mathematical realism and naturalism?
Re: the latter.. If they think of the supernatural as gods and ghosts, as most people to, then there isn’t because mathematical realism doesn’t entail anything like that. I think the ghosts and ghoulies definition is what people care about.
“No , because a Tegmark level IV mathematical Universe—where the apparent physical universe is just a small.part of the mathematical one—isn’t obviously contradictory. (It might be better to say physical objects aren’t inherently non mathematical).”
And a point here not mentioned is your comment is that these mathematical objects if coupled to a normal universe still exhibit different properties: they don’t have a location, they are timeless, they don’t exhibit many of the usual attributes of physical objects etc.
The statement of physical objects not inherently being non-mathematical is a good one—I like that. Clearly there is a deep link between mathematics and nature as the mathematical attributes get exhibited in many places.
“To argue for realism against anti realism, you need to show that “ansatz” can’t work under anti realism, which you haven’t done. If the physical universe is at least partly describable mathematically, , then random guessing at mathematical models will work occssionally. So you dont need to assume more for Anstatz than for UEM”
(Cont’d) apologies, this is taking some time and I will do this in parts as I will run out of time here and there. Bear with me.
“It doesn’t because, as I explained, the existence of a mathematical.universe implies nothing about the mathematical describability of a separate physical universe. (A Tegmark solution does, but you have rejected it!)”
Did you read the paper? It’s not the existence of a mathematical universe that is used to show it, but, given the framework, I use a cardinality argument—so there’s more work and proofs and theorems in the paper—I just summarise the cliff notes in this post for the lay audience. What I do is use a very general statistical observation only, rather than trying to link up individual objects to theorems, I look at the relative size of the spaces. What I have explored is the connection between a separate universe in a framework where the mathematical principles can be used to describe things in many universes, the reverse-Epiphenomenal view. Also, I haven’t totally rejected a Tegmark view, it just needs to be subtly qualified—ie the mathematical parts tied together with the universe can’t be unscoped, one is limited by Cantor’s Theorem here.
“Which result?.If it’s not a solution to UEM, why bring it up?”
The result: Equation (54) in the paper. I brought it up because it was interesting, and I hadn’t seen a cardinality argument before. If by UEM you mean Universal Existing Mathematics, well that’s not what the work is trying to demonstrate. It seems like you intended this work to be about proving UEM, and are frustrated that it doesn’t. I’m confused because that’s not what I set out to do and it’s interesting, but not the topic I am looking at.
“Well, it might, but I don’t quite.see what that has to everything else.”
The reason for my aside, was it was an example of what could be true—I’m trying to say I don’t necessarily hold that the universe is always structured and reasonable, it might not be. What I did was assume it, and show the result. I certainly don’t prove it. It would be interesting to get more data and explore these things. But in the case where the universe is reasonable all the way down, then this would hold. It seems like you wanted me to prove or disprove something which isn’t the topic I set out to do. You can do it if you want. (Cite my paper though :)
“Im trying to find out why you even mentioned UEM. Solips ism?”
I’m really not following you now. I adopt Mathematical Realism for the reasons stated, its a philosophy that’s reasonable and also aligns with the working ethos of physics and mathematics. What other popular flavor should I have chosen?
“I don’t see how it leads to solipsism. “So we actually get a clear solipsism if we go too far with that approach” doesn’t explain it either. It isn’t clea r.”
Ah ok, so the solipsism goes as follows. Note that this does not constitute a proof, it’s really more pointing out a metaphysical convenience, where physicalism on the other hand needs to jump through some more subtle hoops and goes a little against Occam’s Razor. That is: in the specific version of physicalism that seems mathematical entities as nonreal, and fictions that exist in the human mind, then structures that pre-date humanity, like the mathematical groups associated with particle behavior in the early universe, couldn’t’ve existed back then. So I intuit that the actual structure is timeless and not incorporated easily into the universe that way. It’s not the only way, there are other versions of physicalism.
Cont’d (2)
8. “You seem to be blurring “whether mathematical realism is true” and “what are they implications of MR”. If MR is true , then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical.”
I wasn’t able to perceive the blur between these two items in my quote: ”!!!! That’s not at all obvious, in fact, I’d never heard it before until I began looking into this. If it’s obvious now to many people, that’s great! Perhaps the cultural mindset is different compared to my younger days. Certainly the most common view I run into among academics ‘spoken’ is that maths is included in the physical universe as a kind of ‘convenient fiction’ and the physical universe is there is, but then in reality they often kind of tacitly adopted this kind of mathematical realism—I wanted to explore why there’s some unwillingness to face this hypocrisy… (etc)”
I’m simply relating to you what my experience has been in regard to the conversations I have had. The observation that some people will state they adopt a view, and then operate with another view, is not a commentary on whether MR is true or not, or whether I believe it to be true. It’s something that I noticed, and I have in this work taken a stance of MR being true.
I’ve tried to be as clear as I can, and I havent deviated from the same point that I, A) adopt MR (I give some context as to why I think it’s reasonable and a natural view, for my own part), and then B) I develop a formalism assuming it, in the format that I describe (there may be other variants) and then I work through some of the implications of the formalism. I feel like I’m repeating myself over and over to you, I’m not quite sure why it’s not clear.
When you state ‘If MR is true, then the mathematical universe is obviously bigger than the physical universe,just because most maths isn’t physical’ doesn’t follow, in my view. Why would MR being true mean that the mathematical universe is necessarily bigger than the physical universe? The only way I can see it being obvious is if you are defining the ‘mathematical universe’ as the physical universe and the mathematical part altogether. Perhaps that’s what you mean, but you didn’t state it.
In the work I have put together, you can see that what I am showing from the formalism is that extending the physical universe to encompass the mathematical truths runs into some practical issues, and instead defining the universe as being smaller than this, necessarily leaves out some objects that are now not in the universe.
9. “Assuming realism...?” As mentioned, yes the formalism takes MR as a backing metaphysic as previously stated.
10. “The first google match for “Carrier supernatural” is… https://www.richardcarrier.info/archives/7340# ”
Ah, I haven’t read this author before and didn’t realise Carrier was the name of an author, I had assumed it was a jargon coined on here perhaps. But having a brief look, I’m honestly not sure how to match my work to this definition of supernatural. Do I have to? I am anticipating my work can be standalone and not to try to use another person’s definition of supernatural. Here, I use it purely as ‘not in the universe’ where I’ve defined the universe in the way I describe in the work.
I mean, it’s interesting, but one thing that makes it difficult to do matchup with another work is some of the terms don’t appear to be carefully defined enough, e.g. “Tautologically a natural world is a world with nothing supernatural in it, and a supernatural world is a world with at least one supernatural thing in it.” It’s not clear to me what a ‘world’ is, here, or how it is intended to relate to the kinds of realities I am talking about.
11. “Well, maybe aren’t using, or don’t care about the literal definition.”
In that case, umm. I got nothing. If people are using different definitions or don’t care about the literal definition, it doesn’t really impact the meaning of the work I am trying to do here.
12. “Between mathematical realism and mathematical fictionalism , or between mathematical realism and naturalism?
The inherent contradiction I was meaning here was more the former: a Naturalist is bound into believing that the natural laws, mathematical principles governing nature (and so forth) are part of nature or an emergent reality, in some versions, it is present in our mind, and yet with the other hand, will operate as though there was a ‘math land’ where only some items from that apply to our universe. Both valid points of view but incompatible.
In the latter case, you could totally have a Naturalism that extends the physical part to encompass the abstract and mathematical part, as described above, though it need to be done carefully and some methods of doing that can result in contradictions (the ‘draw a box around everything’ scenario). I apologise for the confusion that these are two separate points.
13. “Re: the latter.. If they think of the supernatural as gods and ghosts, as most people to, then there isn’t because mathematical realism doesn’t entail anything like that. I think the ghosts and ghoulies definition is what people care about.”
I hear you, but then that mixes in some folklore aspects adding another dimension of complexity. In this post (ie not the article, but the post) what I noted was that this opens the door—once you admit one supernatural supervening, it demonstrates at least one case where it can occur. I’m of the view that the folklorish aspects in humanity don’t come from nothing, and while many of the folklore stories may not be true, they keep coming up in every culture. I’ve been thinking for a while that the difference between a ‘demon’ and a ‘mindset’ seems slight, and there might be some truth to the idea that the abstract has a more ‘real’ aspect and dimension than people are in the habit of believing right now. That the mind is participating in real, genuine discovery and creation when it deals with mathematics.
And also, it doesn’t matter what people think or care about, let’s work out the truth first, and then we need to believe it, regardless of how uncomfortable it is, or what previous propaganda says, or it has a bad reminding taste of some folklore. So many weird physics things that seem unbelievable and seem crazy I have had to accept over the years as truth. If it’s true it’s true, and I think these aspects are also talked about in the neurology book The Master and His Emissary by Iain McGilchrist and his follow up book ‘The Matter with Things’, which argues that both the ‘Reality-Out-There’ and the ‘Made-Up-Miraculously-By-Our-Minds’ views of reality are both false, and that there is a contribution from both the observer and the environment at the same time in creating reality. The reductionist view of nature to boil it down to something more objective, you can see, is actually highly stylized, attitude driven and not objective at all.
Because most maths isn’t physically applicable, as I stated, and you agreed.
You have a communication issue, because you are not using “supernatural” in the expected way, and a PR issue, because a lot of your intended audience are going to reject the supernatural out of hand. Whence the downvoting.
You need to communicate clearly , and you don’t need to repell the reader
Again, that’s not the same thing. The existence of X-ishly describable entities doesn’t imply the existence of free-standing X’s. For instance, we can describe the colours of external objects using the trichromic RGB system , but it’s definitely not out there.
Comment 1: “Because most maths isn’t physically applicable, as I stated, and you agreed.”
Response 1: I do agree that most maths isn’t physically applicable, but that doesn’t mean that for MR, the MU is obviously bigger (to clarify, for MU here, do you mean physical+maths, I am assuming not). For example, I might have many physical objects in my universe, and not all being mapped to by a mathematical abstraction. I have no way of ensuring that the universe is all totally mapped to. I make a supposition that in physics, we hold a view that it can (or should be). But I don’t know for sure, and so the relative sizes of physical + mathematical parts is hard to define. It may be the case the the mathematical part is indeed larger, but the fact that most maths isn’t physical doesn’t guarantee it, it would be something to do with limits on the size of the physical universe, and/or the scope of mathematics obtruding into it. Maybe most physical doesn’t get mapped to (though, I don’t believe that currently, it could definitely be proposed).
Comment 2: “You have a communication issue, because you are not using “supernatural” in the expected way, and a PR issue, because a lot of your intended audience are going to reject the supernatural out of hand. Whence the downvoting.”
Response 2: Thank you for the view. The way I see it though, it is actually a good communication method, in that I have excited some commentary and engagement from the community—such as yourself—you have been very generous with your engagement. The rejection out-of-hand though accidentally demonstrates that the audience might not have been as attentive as they might pride themselves on, however, which itself is a useful insight to note.
Comment 3: “You need to communicate clearly , and you don’t need to repell the reader”
Response 3: I am attempting to communicate as best I can, and am limited of course by my competence. Apologies if it doesn’t come up to scratch—I am doing my best. I also am not intending to repel the reader, but get some engagement, which was successful.
Also, the ‘do I have to’ was in the context of whether i need to match my work to this definition of supernatural, not based on communicating clearly, per se. I wasn’t aware of the work, but how else do I generate discussion to get some improvement from the lesswrong community? I have to start somewhere. I was as clear as my faculties allow. I tried to define the supernatural the way I see it. A comparison of that view and another work seems like a different topic beyond the scope of this post.
Comment 4: Again, that’s not the same thing. The existence of X-ishly describable entities doesn’t imply the existence of free-standing X’s. For instance, we can describe the colours of external objects using the trichromic RGB system , but it’s definitely not out there.
Response 4: I didn’t say that the existence of X-describable entities implies free standing X’s. The idea of free standing X’s, ie a kind of Platonism is not something i set about to prove. I believe I assumed it as a starting point, and wanted to see how far I could get with it, as an exercise. So I wouldn’t argue it that way. I do state in the above quote that the Naturalist was bound to believing natural laws (by definition) and that I take it to mean that mathematical principles governing nature emerge from this same (physical) universe, as opposed to a more Platonic view of Mathematical Realism (ie. “out there” as you put it). Is that untrue? In regard to such a Platonic view, I would take it that the colours of objects using an RGB system, which are both concepts (the colours, and the RGB system) are abstractions that have an existence, and would be included as one of the abstractions in my formalism, that then get projected down in a reverse-epiphenomenal way, onto the physical-world object. (ie they are attributes of it, and attributes are abstractions).
I have. There’s almost nothing to it, if Ansatz means nothing more than some guessed-at mathematical descriptions turning out to be right.. and that is the . description of Ansaztz you give here:-
(We don’t know where Ansatze come from in a detailed way, but it’s hard to see why that would need a supernatural/metaphsyical explanation, since we don’t know where “think of a number comes from”, but don’;t doubt that it is an ordinary psychological process. The whole rhetoric surrounding Ansatz, or guessing as I like to call it, is overblown, IMO).
But Wigner brings in further issues—the issue that a guessed-at mathmaticlal structure which is intended that is intended to describe one phenomenon, can be applicable to others. And you mention , in relation to Dirac’s relativistc wave equation, the ability to make successful novel predictions..
The various sub-problems have various possible solutions.
An ontology where the universe is based on a set of small set of rules explains the unreasonable effectiveness well enough: since each rule has to cover a lot of ground, each rule has multiple applications. And such an ontology is already fairly standard.
There’s also an underlying problem that saying “I can solve X” has two meanings: “My assumptions are the only solutions to X” and “I have the latest in a long line of putative solutions” It is not enough to succeed, others must fail.
Responses:
Comment 1: “I have. There’s almost nothing to it, if Ansatz means nothing more than some guessed-at mathematical descriptions turning out to be right.. and that is the . description of Ansaztz you give here:- (etc) ..then, so long as some things are mathematically describable, some mathematical descriptions will describe them, even if guessed randomly”
Response 1: Hang on, what I mean is, constructing an Ansatz completely from scratch, without any assumed structure doesn’t sound like something there would be ‘nothing to it’ - I would expect that if you have one, you’d need to be careful not to accidentally smuggle in an assumed concept from the get-go, which hasn’t been demonstrated yet—it’s hard to come to any logical machinery or systems from scratch without assuming something, without any structure or rules or symbols at all. Even if it is very simple, you have to start from somewhere. I tried to keep mine very general, and a few items of structure were added as minimally as possible. But what I mean is, the very concept of an Ansatz itself is automatically couched in some framework—I don’t think one can have a concept unless one at least has a framework for the concept to be part of, or to exist in, so I would assume to even invoke the concept, a framework (even a skeleton one) has been assumed.
Comment 2: “(We don’t know where Ansatze come from in a detailed way, but it’s hard to see why that would need a supernatural/metaphsyical explanation, since we don’t know where “think of a number comes from”, but don’;t doubt that it is an ordinary psychological process. The whole rhetoric surrounding Ansatz, or guessing as I like to call it, is overblown, IMO).”
Response 2: In terms of where Ansatze come from, I don’t think we do know quite where it comes from but we don’t need to know for the purpose of this investigation yet—it’s simply enough that we require logic to exist, and for there to be an abstract concept that can be invoked—very little else in the proof was assumed. The supernatural explanation (again, being careful to define what I mean here by supernatural, that is being outside the physical universe) comes about naturally, with only some minimal rules of logic being invoked. We might not know ‘where’ “think of a number” comes from, but we do know that the number is consistently definable, it ‘exists’ (that’s taken based on an MR viewpoint though), and it gets instantiated a lot, in the physical universe—ie the physical objects obey (and have an intimate relationship with) these mathematical objects.
In terms of the psychological process by which we access it, the psychology would be developed from brain structures, and those are based on proteins, based on info from genes, on chemistry, on physics, down to the smallest particle, so at every level, we have seen a great deal of natural processes are respecting mathematics, and we can write down these laws. So it would come at no surprise that our brains are also structured and follow processes. But, you wouldn’t argue that if the brain was destroyed, that the concepts being referred to by some maths would be destroyed, nor would an atom being destroyed mean the concepts of mathematical groups an equations of motion would be destroyed. Surely those are just all instances but not the thing being referred to itself. (ie they are not the mathematical truths themselves, as those truths turn up in all sorts of places).
Apologies if the rhetoric seems overblown—can you specify in what way? As above, I haven’t quite got your view in mind re mathematical truths. It seems you can’t have no metaphysic, we all have a metaphysic in mind, just it might be undeclared or unexamined—so I am interested to learn yours—it seems yours, to you, seems preferable, but I am unclear of your statement of it.
Comment 3: “But Wigner brings in further issues—the issue that a guessed-at mathmaticlal structure which is intended that is intended to describe one phenomenon, can be applicable to others. And you mention , in relation to Dirac’s relativistc wave equation, the ability to make successful novel predictions.. The various sub-problems have various possible solutions. ”
Response 3: What are the issues raised by Wigner issues for? ie it seems consistent with the metaphysic I have adopted. Many different mathematical mechanisms can be applied to describe processes. It happens all the time in particle physics. There’s a concept of a ‘Representation’ of a group. Subatomic particles are arranged into Groups, which have a certain mathematical structure. But a group is quite general, and you can represent a group in different ways. One particular group might have many different representations, one using matrices, one using complex exponentials, all sorts. These representations have the group structure, but they add more, adding specificity, and are called vector spaces. You could use one machinery to look into a physical phenomenon, or you could use another. Both could apply. Hence why it is hard to have a prescription for how to ‘select just the mathematical truths’ applicable to the physical universe so as to bolt them on and get a consistent P+M Universe (hence why I don’t go down that route).
Comment 4: “An ontology where the universe is based on a set of small set of rules explains the unreasonable effectiveness well enough: since each rule has to cover a lot of ground, each rule has multiple applications. And such an ontology is already fairly standard.”
Response 4: The universe being ‘based’ on a set of rules is an interesting phrase, as it does seem similar to my version of MR—ie that there are rules, and those can be talked about in a meta way, regardless of physical universe, and then the physical universe can be talked about as following those rules. I also agree, since it is the view I was expounding, that this leads to an explanation of the unreasonable effectiveness, but the way I said it was different—I just counted the countably-infinite number of possible abstractions that could apply to a phenomena in a physical universe, and the seemingly ‘smaller’ (more restricted) countably-infinite number of abstractions applying to phenomena that also are extant in a universe, and found them to be of the same Cantor cardinality.
Comment 5: “There’s also an underlying problem that saying “I can solve X” has two meanings: “My assumptions are the only solutions to X” and “I have the latest in a long line of putative solutions” It is not enough to succeed, others must fail.”
Response 5: It is true that a phrase like ‘I can solve X’ can have an ambiguity. I take it to mean the latter though, in other words, taking some assumptions, and working through some steps, one can arrive at a valid solution—which in and of itself has merit, without stating whether other approaches can get to the same point. One might point out the neatness (less ‘epicycles’) or more (or less) in line with Occam’s Razor to evaluate a purported solution after it is given—but I wouldn’t say that means a solution hasn’t been given. Certainly in my work I don’t state it’s the only way, it’s much smaller than that, as a claim—just that this way seems to hold up, seems neat, a lot of things fall out of it ‘for free’ (ie it solves the problem with low entropy, without over-engineering extra unnecessaries) and it also seems to align with thinking drawn from multiple different fields of knowledge which is usually a detective’s ‘hint’ in the right direction, during an investigation.