Reflexivity of the accessibility relation just says that the actual world (whatever that is) is always a possible world (whatever that is). So:

So a rock having the potential energy to fall can be said to be possible within that world itself.

yes of course. The possible worlds of various counterfactual states of the falling rock are generated by how it could fall. Collapsing the sense of that could to simply be worlds which differ counterfactually gives only an extensional definition of the could without the means that the terms in that extensional definition were paired to it.

You could say the same of any possible world, actuality becomes just an indexical property if its sense is equated with the reflexivity of an accessibility relation.

There's a rejoinder I've seen before about a hierarchy of possibility senses:

physical possibility < metaphysical possibility < logical possibility

I think the OP's operating from a perspective from where giving an account of physical possibility is a healthy part of the metaphysics of this world rather than treating 'the metaphysical', whatever that is, as some indeterminate excess of the physical (whatever that is) - with usual examples of 'laws with changed physical constants' or (arguably) p-zombies.

Isn't it more interesting to try to give a good account of how potential and actuality relate as concepts and realities than a deflationary account of both in terms of possible world semantics?

I don't think the OP is pointing out anything particularly deep. Once you fry an egg the raw egg can't be gotten from it. Similarly, you can't make a fried egg sandwich without frying the egg. Obvious check towards @apokrisis here regarding dichotomies and constraints and the arrow of time induced by irreversible processes.

You could of course model that with a poset accessibility relation producing a linearly ordered series of worlds; like a flow chart for making a fried egg sandwich. But...




I read that the difference between the adjacent possibility in the OP and the adjacent possible being the nearest neighbours of an index world in the graph of an accessibility relation is that the possible is internalised to the world. It's the difference between 'this rock falls in some possible world' as a sense of possibility for the rock falling vs 'this rock has the potential energy to fall'. More precisely, the sense of possibility potential energy imbues to the rock vs the sense of possibility of it starting to fall in a nearby possible world.

I think there's also an implication that the potential-sense of possibility for rock falling is logically prior to the possible-world sense of possibility for rock falling, the latter would be said on the basis of the former. Adjacent possibility (potential being constrained by the actual) being the condition for the possibility (lulz) of substantive possibility (contingent truth or falsehood turning on holding in a possible world).

Maybe it's easier to see why this works if you substitute in P is 'this shape is a cube', Q is 'this shape is a cuboid' and R is 'this shape has at least 2 equal sides'. This set of P,Q,R satisfies P=>Q=>R

It's true that if you have a cube (P), then (=>) it's also a cuboid (Q) (this is P=>Q). So (=>) it's true that you have at least two equal sides (R) since you have a cuboid (P) (this is P=>R).

Another way of thinking about it is that implications are machines that take inputs and give you outputs. If P is true on the left as well as (P=>Q =>R), that means that if you have P and P=>Q, you get Q, but if you have P=>Q, you get P=>R since Q=>R.

It's a lot of conditionals, but the main thing that you have to notice is that you want to assume everything on the left of the =>, then show that assuming P=>Q then assuming P allows you to derive P=>R, that then means since you derived P=>R from assuming P=>Q you get (P=>Q) => (P=>R).

Chain of implication proofs generally proceed by assuming things on the left, then leveraging the assumption of the implications to get the things on the right, then since you assumed the things on the left they imply the things on the right.

This is pretty lucid and far more detailed than my exegesis. It's also mostly jargon free (things are expressed outside of Heideggerese).

Heidegger's central commitments throughout his work, at least as I see it (which should be taken with a pinch of salt), are the relationship of humans to being/the world and of truth to humans and language/the world. It shouldn't come as a surprise, then, that Heidegger sees truth and untruth as ways of relating to being. Heidegger's use of poetry and poetic language is adopted largely because he sees it as a way of relating to his topics of interest in a manner which exhibits them well.

Heidegger accuses most philosophy since Plato as thinking of truth as the correspondence of matter to language about it. Like 'the cat is on the mat' is true if and only if the cat is on the mat. This can be termed adequation or likening; when the words are determining the thing through truth and the thing is determining the words as their accompanying state of affairs.

Heidegger prefers to think of truth as unconcealment or illumination. A good example here is that of a statue sculptor. The sculptor has a block of stone to work with, it has some properties and shapes which the sculptor can use to work with the block to bring a shape about. If the sculptor chooses to sculpt in way X, she cannot choose to sculpt in way Y; once the marks are made the shape arrives.

This is similar to how Heidegger thinks of truth. Bringing out the nature of a topic by relating to it adequately. But also, relating to it in some way always curtails relating to it in other ways. As things are uncovered by adopting a framing device, so too are some things concealed. Illumination also casts shadows.

Since he sees most philosophy since (and including) Plato as an obfuscation of the truth of things and a forgetting of the question of being, it's quite natural to try to adopt a different method which illuminates in the right way and casts shadows over the irrelevances.* He sees this in poetic language; it's very expressive, metaphors function similarly to plain language, suggestiveness is incorporated explicitly in how line follows line. It is use of language without annoying methodological constraints on expression.

Heidegger sees this use of language and its accompanying orientation towards illumination as very fundamental and basic, I agree with him here. As an example; consider as a model of truth rather than the statement like 'the cat is on the matter', the exegesis of an idea. Does how illuminating my exegesis of Heidegger is turn on the truth of the statements in it? Somewhat, certainly, but more importantly it's the use of truths to paint a picture of Heidegger's position on poetry. What matters is how readable it is, how accessible it is, whether the examples are good, whether the terminology I adopt is good and so on.

I think I'd achieve a worse exegesis if I wrote things like:

'poesis is close to the originary operation of alaethia in language, in some respects poesis is the use of language aimed at the hermeneutical rather than propositional as-structure; where proposition is the adequation of a concept or statement with its object'

despite it being a more precise description using the right words. :P

edit: * his views on technology and his nationalism about the German language slot in about there.

Heidegger's pretty hard to skim read. You really have to pay attention to how the sentences follow from each other until you get used to how he thinks and writes. He's also expressedly not a humanist (see the 'Letter on Humanism') like you stated in the OP.

Related but distinct. Heidegger is quite silent on ethics, even though he makes use of normativity in his account of making sense of the world.

Technology is treated as an obstacle to forming a deeper understanding of being. This is because it is a way of interpreting being. A reasonable analogy here is being asked to give an account of sight if everyone wore blindfolds all the time and weren't aware of it; you'd have to remove the blindfolds before the account could be made.

The Heidggerian move is: since technology is one way of relating to being and, as a framing device, invites us to view things in some ways - like as goods, resources and instruments - it must be dependent on a less conditioned sense in which we relate to being. This fundamental sense is that being itself is a concern for humans. Humans can't help but care about it.

In the every day life, Heidegger attempts to show how this care for being; which is intimately linked with intentionality; generates contextualised interpretations of our environment that automatically prescribes what it means to be in that environment. Like the sense of being at home while relaxing there, or of being in good company when with a close friend.

In terms of philosophy, this care is orientation towards a problem whose contours are circumscribed by adopting sufficiently precise framing devices and sufficiently illuminating examples to study the problem. This might not seem like a particularly advantageous methodology for doing metaphysics with, but it certainly suits a type of being (humans) who is always concerned with being. What is the problem? The problem that we are always concerned with, being... This iterative refinement of framing devices zooming in on a series of conceptual presuppositions is called the hermeneutic circle, and attempting to describe things in a way to exhibit their conceptual presuppositions is called formal indication.

Technology is an obstacle for that method, as it becomes difficult to see past the constraints it places on developing an ontology; questions of how and for what applied to specific entities, properties of assemblages of entities. Inquiry guided by 'how' and 'what for' questions can never raise the question of the being of those entities; the existence they share in.

A Cliffnotes summary with no academic references of Heidegger's view on technology.

Heidegger cares a lot about the way people examine things. He cares a lot about how problems are framed; how questions are asked. Specifically, he thinks ways of asking questions can be occlusive of the subject matter. The easiest example of this is how Heidegger differs methodologically from Husserl.

For Heidegger, Husserl's phenomenology starts in terms of contemplating perceptual objects. Like rotating an apple in your head or seeing the colour of something; from considering those kind of things, Husserl attempts to draw out the essential nature of experience. One example of this essential nature of experience Husserl highlights is intentionality. Consciousness is always consciousness of something; it is directed towards a specific object or task.

Heidegger thinks this is a bit wrongheaded. Specifically, he questions the way Husserl is arriving at answers; should the essential nature of experience be highlighted through abstract mental operations, or by attempting to draw out the essential character of how people experience stuff on a day to day basis? Heidegger concludes (with a lot more words and nuance) that intentionality isn't just direction towards a specific object or task, but of being in a situation.

Say Eric Clapton is freestyling on his guitar, he's damn good at it, he's performing and composing at the same time, it's badass. This is formed from an interplay of attending to how the situation was (the previous bit of improv), how the situation is (the previous bit of improv being extended by playing appropriate notes) and how the situation could and should develop (expressing whatever theme Clapton is experimenting with). It's also an awareness that has a sense of normality to it - if one of his strings breaks the situation changes. This is a very open ended sense of intentionality; lots of things and relations of things are attended to when Clapton's playing the guitar.

So, Heidegger's derived a different sense of intentionality by asking how humans do purposeful activity vs underpinning constancies in how humans perceive objects. This is to say, he approached the question in a different way.

Heidegger thinks technology, fundamentally, operates on a similar level to intentionality. It's a banality to say that technological advance has changed how people relate to each other; like Tindr, Facebook, Twitter and even the phone. Heidegger wants to reveal what underpins this banality - in what ways has technology changed how people approach doing every day tasks, and what conceptual structure does this change in approach have?

A pretty good analogy, in my book, is that Heidegger thinks technology is like Tindr. Romantic partners are reduced to a standing reserve of sexual partners, and people on Tindr are reduced to their attractiveness for the person; that is, people have become instrumentalised; put to work, appraised in usefulness with regard to a task.

Tindr is a model of instrumental intentionality applied to people. Technology, for Heidegger, is fundamentally a way of seeing things in terms of their usefulness. Apply this to the entire world, and you get the start of the homo economicus myth: the world is a competition over finite resources.

Technology, then, is this attitude applied everywhere. Things start to look like goods and resources. Again, it's pretty banal that most things that we encounter on a day to day basis can be seen as goods and resources; but that's precisely the point Heidegger is raising! Technology is an instrumental framing of our relationship to the world, one in which we see people as subjects for resource allocation (or as 'cogs in the machine').

'The Question Concerning Technology', 'The Age of the World Picture' and 'The Origin of the Work of Art', are three essays of Heidegger that deal with the different parts of this account.

The Question Concerning Technology sets up this account of instrumental rationality.

The Age of the World Picture deals a bit with instrumental intentionality as a conceptual foundation for a scientific worldview.

The Origin of the Work of Art deals with how Heidegger wants to get around the constraints of a technological worldview for a more 'fundamental' interpretation of our relationship with the world (of which instrumentality is one possibility).

The introductions to Being and Time deal with his methodology more generally (even though it arguably changes a bit in his later works). I think they situate in what register Heidegger is asking questions pretty well (but they're very dense, especially the second one).

C'mon man, this is just pedantic. Fair's the best model a priori as it's a pretty good approximation to the base model odds of heads vs tails close to 1. You need relatively a lot of data (long sequences of flips) to distinguish a generic coin from a perfectly fair one, even though irregularities will probably make any given coin slightly non-fair. This is 'cos fair coins produce maximally variant flips.

You can't even claim 'the coin is unfair' from typical short sequences of flips that are how coin flips are used.

The truth, here, is almost useless.

Yeah. But most coins are not loaded. And many coins are flipped!

Man who upon only seeing 12 heads in a row still assumes fair coin is silly man. Man who assumes it's a double header upon only seeing 12 heads is almost as silly.

Cakes (genus) food (species)

A non-food must not be a cake, since if a non-food was cake it would indeed be food. This is because cakes are foods. Another way of saying that cakes are foods is that cakes are a genus of the species foods.

A non-cake could still be food, for example a sandwich. This is because some non-cakes are still foods.

So there are more things which are 'non-cakes' than there are things which are 'non-foods'.

There's a similarity between a good proof and a perfect example. If you find a really good example of a structure; good meaning it exhibits all or almost all the moving parts of the general thing in a more understandable or otherwise easier way; it becomes much easier to deal with the general thing. There's a few youtube channels that are devoted to doing this (3blue1brown is the best IMO), and they're incredibly satisfying to watch if you have some of the mathematical background required.

Maths in this thread is serving as an example like that, I think. It's illustrative of the general structure of reasoning in lots of important ways, and also illustrative of how institutional study influences things. Like the relationships of conjectures (problems) to research programs (developing germinal methods or applying them to other things).

Applied maths and stats don't have the same relationship with conjectures, or rather the conjectures change. They're often less formalised intuitions that things will work out in some nice way when developing method X or applying method X to data or physical process Y. But you do get good examples (for certain problemscapes) in the sense I discussed above.

That was my reading of the article too, yeah. In a different context the proof might look very deep.

I agree. I don't see how either of us could conceive of the concept of 'hinges' without being able to abstract them from their use. — Pseudonym

In my view this statement is pretty much the point of the thread, the math thing was just illustrating this point. What do you think happens to the idea of a 'solution' of a philosophical problem when what counts as a solution depends upon a framing device? This isn't rhetorical.

With regards to math, mathematical conjectures (problems) and research programs (framing devices carving up problemscapes) are just as important, if not more so, than the canon of mathematics. I think this is demonstrated by why the recent Abel Prize winner won the prize: not just what he proved, but the way he proved stuff and the exciting conjectures his methodological development allowed,

I don't really think there are hinge propositions, but I adopted the vocabulary because it was appropriate. Hinge propositions, as used in philosophy, are a gloss of certainty on a statement of incommensurability. They function to make one way of analysing, or one set of beliefs, be fundamentally at odds with another. I'd say they're a bad model of necessary presuppositions for doing something, because they have an inappropriate sense of necessity.

If you look at the history of philosophy, there are a lot of methodological innovations; and the 'great thinkers' transform(ed) how philosophy was done, not just what positions were adopted. Philosophy is absolutely destructive to all presuppositions, it doesn't just question it questions questions.

Not at all. I'm arguing, as Salvatore does in response to Wright's 'hinges', that one cannot simultaneously allow the discourse to range over both the argument given the use of hinges and the selection of hinges, which is what happens in philosophical debate. To do so would be a misuse of the terms used to judge positions in ordinary use. The selection of hinges is mere feeling, that's the point, if they were not then the problem of 'selecting hinges' would itself require hinges in order to resolve it, and so on ad infinitum. This is what I meant originally by 'having your cake and eating it'. I entirely agree that maths may well proceed by certain selections which then dictate the nature of the solution (I'm hardly in a position to disagree, given my mathematical knowledge), but the fact that philosophy will also proceed thus is trivially true. All that's being said there is that the selection of axioms/hinges/problmescapes will constrain the set of possible solutions. Solutions are constrained by factors (which themselves must be presumed to be true). I don't see anything controversial there. — Pseudonym

Can hinges be analysed in contexts in which they are not presumed? It isn't as if everything that's required to philosophise about X is required to philosophise about Y. I take it that we're actually doing this at the minute; we're arguing about the framing of philosophy itself. Each of us is using a different framing device. This is supposed to be an impossibility, but it's not. I have an idea of what philosophy looks like under your frame, and it doesn't look like philosophy to me. And vice versa for you. To operate on this level of abstraction has as a hinge that we can take other hinges and philosophise about them.

Philosophy is different in that the psychological effect, the human reception, of the solution actually matters, matters way more than the constraints (which are trivially surmounted by simply re-arranging axioms), matters way more than the signposts of previous thinkers (which can be discarded as easily as logical positivism). What we're producing as a solution is an attractive salve to the wounds caused by the uncertainty of that which is as yet unknown. Look at the continuing popularity of the Cosmological Argument, the staying power of 'Meditations'. Do you honestly think these are solutions which just fell out inevitably from the definition of the problemscape. They are crafted such as to make the water flow where its needed.

I'm seeing something like the idea of hinge propositions in what you're writing. Hinge propositions are certainties required to partake in a discourse. It's very easy to elevate something to the status of a hinge proposition when analysing a discourse that makes use of it. It'd be perverse to do theology without some divine, comparative theology without different divinities with common concepts, platonic ontology without form and instantiation and so on. To my mind, you're characterising the adoption of hinge propositions as a kind of psychological excess to the discourse; why engage in this rather than that? Must be mere feeling.

Try going one level up in abstraction, where instead of agents with desires adopting hinge propositions you see adopting hinge propositions as opening the door to partially formed space of problems. The truth or falsity of such propositions isn't really their point, what matters is that they are held certain; beliefs in them are shown in actions. Transpose the hinge propositions from psychological defence mechanisms to the logical register they function in: preconditions for partaking in different discourses.

Go up one more level of abstraction; where do the preconditions come from? At this point you could collapse this chain of abstractions again into the individual; preconditions are adopted as defence mechanisms of worldview. What I'm trying to say is something like: the behaviour of these hinge propositions has its own dynamical character. You can chart the adoption of hinge propositions as moving through gateways to further discourse; at this level of abstraction hinge propositions don't look much like propositions, they look like framing devices.

Go up one more level of abstraction, where do the framing devices come from? At this point, you could collapse this chain of abstractions again into the individual: framing devices are adopted as defence mechanisms of worldview generation. The behaviour of these framing devices has its own dynamical character. You can chart the adoption of framing devices as moving through gateways to further discourse: at this level of abstraction, framing devices don't look much like worldviews... What do they look like, though? Does whatever rootedness in people and discourse they have do anything to determine their character? Yeah, probably, but what do they respond to. What's grist for the mill of framing and adopting discursive constraints? You could say 'it's the person', but that's a category error. Yes, inquiry is something people do, yes thoughts are expressed by people for a variety of reasons, but why respond with thoughts in way X rather than way Y?

I think you stop the analysis at this point, because you've already decided that framing is a function of prediliction and nothing more. So of course it seems that everything reduces to prediliction when you frame things this way; that's all it could've been. There's no evidence or example I could give you which can't be reduced to an externally structured prediliction...

So go up one level of abstraction, what do the predilictions respond to? How are they created? Is what you posited as external to the vertigo of philosophy actually external, or is it more grist to its mill? Do the same logical operation we did when transposing the hinge propositions into discourse. And transpose this responsiveness of philosophical prediliction into philosophical thought. This gives you questions like: what induces insight in philosophical inquiry? What is philosophical intentionality or directedness? Briefly: obtaining a felicitous concept which is oriented toward a problem.

At this point, I imagine you're thinking 'but this isn't philosophy, this is inquiry in general, where is the specificity of philosophical problems?' - and that's kind of the point. There are no specifically philosophical problems; which isn't to say philosophy can only be given a negative characterisation, it's that this exterior directedness is always part of philosophical inquiry. Philosophical novelty is achieved by employing framing devices felicitously to bring new problems into the discourse, express them well, and shed light on old problems.

Now, @csalisbury will interject at this point saying (something like) 'this means in principle philosophy can't provide an account of what is exterior to it now, and you just gotta do it'. The same's true of all inquiry. @apokrisis comes in at this point and then says this is some self contradictory relativist pluralism nonsense, and 'totalising' a problem is the same as demarcating a problemscape (using a different vocabulary); then applies the whole thing to itself again. @StreetlightX comes in and finds dwelling in performative contradiction between 'exterior directedness' (the encounter) and giving a philosophical culmination of it frustrating.

My view is that only a God can save us, really. But luckily God is 'this problem is not within the scope of this paper'.

I expect my question got lost in my rambling prose, but it would help me to understand your line of thought if you could tell me how you think the history of the problem affects the approach. As I said, in philosophy it's absolutely instrumental, I'm not seeing any way it is in maths. — Pseudonym

I think I can only respond to this with a question: what makes you see all the historical details I gave that make sense of the math problem I gave you as unhistorical, or irrelevant historical detail? I see it as something like: we're dealing with codified and canonised mathematical history applied to that problem (integral of $\exp(x^2 /2)$). You don't have to be aware of that history to use its codifications, just like we don't need to know the etymology of a word to use it properly. Which isn't to say the etymology is irrelevant; it traces the history of the word and what made the word what it is.

UK has something similar in the grime scene with Akala, Lowkey and Wretch. But rap as social commentary is old as rap. I think it's more that political rap is becoming more popular in counter culture than rap being some novel transformative force.

Real revolutionaries engage in revolutionary acts, 'far as I know, I just rap — Akala

It's great how you've laid out what you see as a 'problemscape' in maths, that has been helpful, but (and I feel bad I didn’t think to specify this at first) I actually meant to ask what you thought an incomplete problemscape would look like in philosophy. The point being that I'm not sure how such a process would apply in philosophy even though I'm sure it does in maths. — Pseudonym

I certainly know what it looks like better in maths than in philosophy. The post was meant to draw an analogy from maths to philosophy without specifying all the moving parts. Largely because what the moving parts are in philosophy are a lot harder to specify and a lot broader.

I was hoping that the way I described the mathematical example didn't look particularly mathematical; containing the germs of what the corresponding things in philosophy would look like through a blurry-eyes translation from one to the other.

My first step in both types of philosophical problem would be to understand why it's a problem in the first place, I'd first want to know why it needed solving at all, what place they have in the wider problem hierarchy? Already, I'm not sure whether this step is even necessary in your maths problem. Do we need to know why integrals even need solving to approach a solution? I could certainly solve y=x+4 (much more my level) without needing to know why we might need to know X in terms of y, but I wouldn't dream of approaching a philosophical problem without such background. — Pseudonym

What's at stake in your presentation of the idea hopefully captures what's at stake in the problem in general. There's a mathematical procedure which links integrals to solutions, it's a very simple mathematical problem conceptually; you didn't have to come up with the steps or anything they contain on your own, you didn't have to come up with the integral notation, you didn't have to come up with the idea of an integral. All of that's background to the integral problem.

Maybe if I asked you 'find the area under this curve'; giving the bell shape of $\small \exp(-x^2 /2)$... You'd be another Newton or Leibniz and Descartes if you solved it in the manner expected, along with inventing a procedure to match curves to functions eh (a Hermite or a Laplace)? All of that ambiguity was removed from the problem because there are signposts in the problemscape already interpreting what the problem consists of and solution methods. They weren't there to Newton or Leibniz, even if they had the rule of Archimedes (a precursor to calculus) and Cartesian coordinates and functions to springboard from.

I smuggled in so much context with the examples and notation it isn't a wonder at all that you thought it doesn't resemble a philosophical problem. Perhaps the context of it is more clear now, as well as what removing it would give rise to as a problem (reinventing at least a century of math).

Since you mentioned ethics, let's take an alternate history of the utility monster. Imagine that you don't know anything about utilitarianism, and instead are asked 'is it right for all but one person to be in heaven, because that one person is unjustly in hell?'. Say you see a generality in the problem, it doesn't need the religious trappings. So the question could be refined to: 'is it right for all but one person to be in a blissful state because that one person is in abject, inescapable suffering?'. Or maybe the reverse; 'is it right that one person is in utmost bliss because the many are suffering?'

Then you'd probably need some justification for deciding whether it's right or not. Maybe you start trying to compare suffering and bliss on a scale; maybe one ice cream looks like it's worth a punch in the arm. So you imagine on a scale all these people being in the positive side a given amount, and that one person on the negative side a given amount and start to weigh the amounts... By that point you'd have invented a germinal utility calculus. Then you get the weird idea that, hey, what if one person's sensations are super extreme and they derive more bliss from things than others - a lot more -, do you then need to start centralising the distribution of happiness on the person, giving them far more than others, just to get the most happiness, additively, from the scale?

I wanna take the above as a paradigmatic, but oversimplified, set up of a problemscape in philosophy. You start having to do things like come up with conceptual machinery to compare gains against losses; happiness against suffering; implicit in this is an idea of the ethical decision being about the eventualities of your actions (resultant happiness/suffering... utility) rather than anything inherent to the action.

Then someone comes along from a different set of concepts and gives you a deontological response; 'firstly, it's wrong to centralise happiness like that because you couldn't, in principle, centralise happiness for everyone. That's a contradiction in terms. Secondly, I don't want to grant even the framework you're considering ethics; you're doing this stupid scale thing where suffering and pain are traded off against each other as if they, too, weren't part of actions.'

Then the proto-utilitarian invents 'biting the bullet', and discourse stops. Two irreconcilable, in native terms, frameworks about ethics which also disagree on an ethical dilemma.

On a meta-level, a lot had to be in place, in the background, to recognise this alternate histories as images of philosophical debates. The problemscape for setting up the problemscape is looking at a methodology for specifying methodology; a snake biting its tail that nevertheless must begin and end somewhere; the unbounded space of methodological considerations congealed and constrained through the slowly evolving background that makes sense of it retroactively. Inquiry occupies a liminal space between the already structured and the structuring of what is now already there.

So, I don't think the approach I was taking was specifically addressing philosophy, I was trying to get at the general structure of inquiry of which, I assume, philosophy is obviously a part of. Perhaps another way of putting it; philosophy exists as a stratum of ideas and their embodiment in studying philosophy; the ideas modify the study, the study modifies the ideas.

Edit: if any of this seems somewhat trivial, good. I hope, then, it is now trivially true of philosophy.

First, I'd ask what an un-set-up problemscape looks like. In order that some job of work needs to be done to set one up, I think it's reasonable that you should be able describe an unfinished one. Second, you say "allow" the problemscape to be navigated. I'll skip over "navigated" for now lest you literally start tearing your hair out, but "allow" intrigues me. Again, by the same method, what would an approach which did not "allow" navigation look like, how would we know we were engaged in such a method? — Pseudonym

You said you're unfamiliar with math. An un set up problemscape for you might be solving this integral:

$\int\limits_{-\infty}^{\infty} e^{\frac{x^2}{2}} dx$

and I can give you a couple of hints to its usual solution:

• you can't solve the integral through standard methods (no plug and play algorithms)
• you need to see the integral as equal to a 2 dimensional integral
• you need to learn something about the transformation of Cartesian to Polar coordinates

I think if I gave that integral and the hints to some math undergrads with university level calculus experience on a homework assignment, a good chunk of them would be able to solve it. In less than an hour. With no Wolfram Alpha. An un-setup problemspace, to you, might be me just giving you that integral to compute with no hints. It has a certain impenetrability, you need to spend some time on the contours of the problem before finding a way in. When you're about the solve the problem, if you're anything like me, you'll be like 'oh god... I can do this*? This will actually work? Holy shit that's cool'. By * I mean 'I can use some details of bullet points 2 and 3 to solve it? Wow.'.

Of course, you could google that integral if you know how to use Google to search equations (standard internet equation notation), then you'd find the standard solution to it. You'll discover that the problemscape for that problem is very well studied, and there are lots of avenues of study leading to and from it.

I imagine whenever you post something on here, you have an opinion on the topic. If the topic's something you've studied before and you're regurgitating pre-articulated thoughts it doesn't require much thinking except in how to communicate what you've already thought. You know your way around the problemscape; the imaginative background you have for the problem; and all that remains is to beam it into your readers' minds with sufficiently good writing.

I don't think you're doing that here, you seem to be in a similar discursive position to me, you've got some intuition on the structure of philosophical problems in general, you see what I'm saying as a flawed instance of that structure (at least on the forum), and you're struggling to find the words to articulate in response to what I'm saying to communicate a disagreement or difference in emphasis. [You've got to wrap up what you think into a neat package; which perhaps can be identified with thinking about the problems in the thread and in your head; then translate the package into a common language.] The stuff in the [] seems to occur all at once.

If you relax the assumption that we're 'beaming things into eachothers' heads' with language use, and instead inhabiting a vast terrain of communal ideas, we get a more interesting location for the problemscape. To be sure, it's something we're doing, but it's also just as much something that's already been touched on by countless others in different contexts. Sure, what we're doing isn't going to be inscribed in the canon of philosophical discourse since we're on a forum, but that doesn't matter that much. It's like we're both on a knitting forum and sharing patterns.

So, I resist your claim what we're doing is mostly driven by psychological peculiarities - we're navigating some abstract space of problems that we share but have our own different copies of. I see the claim that we're driven by psychological peculiarities as misguided: it's like saying the knitter is driven by psychological peculiarities; sure, they are in a sense, but they're also following a pattern or making one up.

I'd like it if one of our few academic philosophers could chime in on their experiences of dealing with philosophical problems. @Pierre-Normand, do you have any input on what it feels like and how you address philosophical problems as an actual academic philosopher?

I'm interested in how you're thinking of 'accuracy' of an idea. If you apply it to the steps of proving a theorem, that's usually pretty banal; people who can read the language the proof's written in will usually be able to see if it's right or not, if the general idea is right etc. There's a derived sense in which a theorem can be accurate; if its proof is. But I don't see that this sense of accuracy applies to mathematical objects or ways of thinking about mathematics in the same way.

There are probably, to borrow from Austin, felicities and infelicities in how theorems are proved and how people think about/imagine mathematical objects. I tried to highlight this with the Dedekind Cuts vs Cauchy sequences thing. Street's article highlights it in a more accurate way; a particularly felicitous way of thinking about the problem lead readily to its solution.

An apocryphal quote (might be real, couldn't find a reference when searching though) attributed to Grothendeick is that 'don't try to prove something until it is obvious'. Similar one by Riemann 'if you give me the theorem I'll give you the proof'. Mathematicians seem to think about creativity in mathematics this way; a certain 'accuracy of ideas' which doesn't immediately reduce to the accuracy of a proof.

Grothendeick really, really thought of stuff this way:

I can illustrate the ... approach (to mathematics) with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.

Thanks!

I'm not gonna communicate with you better than you. But yeah, I think you're onto something. That means I have to stop now, because I can't rehearse it from a script. Speak later. :)

I don't want to play the Socratic authority game and I'm tired. Say what you want to say and let it stand on its own merits.

From a modern perspective, because they provide models of the same object. How they do it differs. That they do it matters, of course, but the really interesting parts about them are how they picture the object. Understanding the latter and the demonstrations of the former are understanding math. Not just demonstrations, which say 'this idea solves the problem, look!', and if you're lucky they solve the problem in a way that illuminates something good about the problem...

That the real numbers come in and provide the 'continuum' for calculus is the overall problemscape. Why they matter as proofs isn't provided by the proofs etc etc etc

Another interesting thread is why the real numbers are called real numbers when to the Pythagoreans they consist of immeasurable bullcrap...

You don't. They both have merits. They're open ended discussions. They're differences in emphasis that make mathematics look different depending on how you view it; since it's a foundational issue. This is like asking 'Are Cauchy Sequences of Rationals or Dedekind Cuts a better way of defining how irrational numbers work?', wrong level of concept. They're slightly different axiomatisations of provably the same thing.

The Cauchy sequence axiomatisation emphasises that real numbers can be arbitrarily approximated by infinite sequences of rational numbers, real numbers being defined as the limit. This is not a definition of a Cauchy sequence.

The Dedekind Cut axiomatisation makes it pretty obvious that real numbers can do weird shit to sequences of rationals; conceptually it's more similar to ideas of why real numbers like e and pi are 'holes' in the rationals despite the rationals getting arbitrarily close to each other. This is not a definition of the Dedekind Cut.

One axiomatisation is suggestive of how to compute real numbers but requires some sophisticated other stuff to get going (how to deal with infinities rigorously). This is Cauchy.

The Dedekind cut axiomatisation is suggestive of the fact that the real numbers as a whole will 'fill in the gaps' of rational numbers by 'plugging an irrational in at each gap'. Like, there's a 'dedekind cut' at the square root of 2 from the rationals which is defined as {the number whose square is 2}. That's basically it. It's very 'mathematician's answer', which is both theoretical simplicity but nonrevealing about much of the connection of the real number line to calculus. One emphasises how to construct real numbers as limits, one emphasises that these limits manifest as holes in the rational number line.

That's how I see it. You might really like Dedekind Cuts, I prefer Cauchy sequences. You get taught both because they're both nice and more naturally fit into some ways of thinking about the numbers intuitively. Also because it's historically the thing to do.

Stahp.

This consensus and branching out of mathematics (what I call "step-wise" fashion) is not possible in philosophy where the constraints of the variables to be discussed are so open-ended. As someone previously brought up, that problems can be framed from a Derridaean or a Russelian perspective would negate this analogy to math. — schopenhauer1

[rant]

Nah man. While you don't see disagreements about whether a proven theorem is true or not (unless it's a proof reading or review of a paper looking for inconsistencies). You absolutely see these open ended disagreements on the relative merits of proofs. Even when you constrain this to pure mathematics (which isn't fair, maths is much broader than pure): Terence Tao talks a lot about using the 'Mellin Transform' to provide deeper or more insightful proofs about the distribution of prime numbers. There are disagreements in pure math over whether category theory or set theory provides a more natural basis for mathematical arguments. Categories can be seen as a generalisation of sets (which they are) or a more fundamental (less constrained) object.

Categories are also a much more general concept than logics, so this 'zooming out to logic', treating it as a court of reason for mathematics, while true in a sense that mathematical proof respects the underlying logic (grammar? syntax?) of its objects, also isn't a good depiction of mathematical meaning. There's a whole other region of mathematical discourse which just isn't captured by the theorems alone; it's about what the theorems depict. The imaginative background. In that realm, intuitions, people following their noses and linking back to the literature holds sway. Even if you want to think about it as a Platonic realm of 'substantive abstractions' you're not paying much attention to how mathematics is actually done; that is, how people relate to and create/discover mathematical entities and relations between them. What was once an idea becomes a theorem. What was once a theorem becomes a research program with some tinkering.

If you wanna see a contemporary example, go to the site 'the nlab' and look at the page on the 'pullback'. The nlab is a bunch of category theorists, and they're making the claim that the pullback provides the meaning of an equation. Even though the pullback as an abstraction was developed in the language of real analysis and differential geometry; pullback measures and tangent bundles. Far less general than categories.

Then, if you wanna actually represent mathematics fairly - you gotta include applied mathematics and statistics of various sorts. Applied mathematicians have the above considerations; elegant representations of differential equations, physically motivated ways of doing calculus on computers with theoretically guaranteed structures of errors. Then there's the whole messy business of actually applying the equations to things. Why do the Navier Stokes equations capture fluid flow? Why is there a debate on whether and how freak waves can be captured by the equations? Those things aren't just Platonic abstractions, there's some kind of embedding to the messiness of physical processes.

Then you go to statistics and you end up with people trying to formalise things like Occam's Razor, conservative inference from data, robustness of conclusions as well as some of the above elements (like theoretical elegance or numerical guarantees). As well as looking at how the fuck to do statistical procedures on a computer (the infinity of the real line or the sample space isn't something a computer can contain y'know, only so many bits.) There's no theorem which will say 'this mathematical concept captures Occam's razor' or 'this mathematical concept contains the idea of drawing tentative conclusions' - all of those things are given mathematical analogues then debated on their philosophical/epistemological, statistical and pragmatic merits.

There's no theorem which says whether a theorem is elegant or worthy of a research program. The Abel prize winner this year didn't just win it because he proved a theorem. He provided a way of linking loads of different shit together in an incredibly profound and elegant way, it made a research program because the ideas had that much merit.

You're completely trivialising mathematics. Stahp.

[/rant]

You don't always need repeated values corresponding to a single parameter to get a sensible estimate for that parameter; look at Gelman's analysis of the radon data...

If you have to redefine the parameters in order to be right, then not only are you wrong, but you are deceiving yourself. That is not philosophy, not by a long shot and if that is the standard that passes on these forums, then I have to question if I belong here at all. — Jeremiah

Oi oi, look in the 'math and motives' thread and see what you think.

I agree with this, for most philosophy though in a practical sense, the definition it really works with is simply "those propositions contained within the canon already labelled philosophy, or those stated by people qualified by their knowledge of such propositions". By which I mean that realistically most philosophers simply let someone else define philosophy for them and only become agitated where there's some suggestions that their current project isn't it. — Pseudonym

In a sense I think this is an essential feature of philosophy, in a sense I think it's enforced by philosophy as an institute. This is getting close to what I was gesturing towards regarding the dissolution of problems; more precisely it's either rendering them irrelevant or positing them as such.

Positing things as irrelevant is pretty easy, but this might speak to my inexperience with philosophy institutionally. If you're doing philosophy within a research paradigm or quite constrained theoretical context then the problems you deal with are prefigured (but not necessarily circumscribed) by that theoretical milieu. PhD student X works in dialethic logic, PhD student Y works in feminist standpoint epistemology, PhD student Z works in mereology. They're dealing with stuff already in a little island of sense; equivalently a frame; with stable ideas of what the problems are.

How they approach those problems might span subfields; like, say, if @Wayfarer was doing work in comparative religion under Graham Priest's framework of interpreting Buddhist logic (four corners stuff) as dialethic logic (which is something I'd love to see a thread on if you're interested/knowledgeable in the intersection btw Wayfarer). But nevertheless there's a prefigured philosophical terrain to navigate. A historical example would be it might not be surprising if you were developing ordinary language philosophy in the wake of Russell and Wittgenstein in Oxbridge in the 1960s. Or if you're there now doing cultural theory with poststructuralist sympathies; questioning 'canons', as I've heard they call it (how times change).

What's a bit harder to shed light on is something like 'all philosophy is philosophy ceteris paribus', when you do it you'll have a certain imaginative background of what the philosophical terrain looks like as you're charting it. This probably requires a bounded terrain; if everything can be relevantly said of an idea it says nothing.

I think that bounded terrain could be meaningfully called a problemscape. Imagine we're in a scenario where we've not applied to study within a specific research program, and we do the far more daring thing of proposing our own thesis to a supervisor (then going through the funding nightmare, assume this happens to :P), and the supervisor actually supervises instead of steers. Provides direction without determination. This implicates that the philosopher navigating the problemscape has to make a few decisions about it. To my mind they've got to do a few things to be doing inquiry in general:

• They have to set up the problemscape somehow.
• They have to provide analytical tools and demonstrations that allow navigating the problemscape; these might be arguments, phenomenologies, references, interpretations of scientific studies etc.
• (1) and (2), despite the enduring myth of the maverick genius, are inherently social. Even mavericks are mavericks because they stand in a certain position towards established knowledge and speak... maverickally... about it; sometimes expand it.

Example, Einstein as a patent clerk was a maverick; but his ideas gained traction not just because they were ultimately more accurate models, but because they provided an interesting perspective with demonstrable links to the Newtonian model while being mathematically and physically more general; which thus allowed inquiring into more specific contexts too; more numerically, more conceptually.

Kripke was probably a maverick when he was developing his approach to modal logic in school; same deal, elegant simplifications and generalizations out the mouth of 'babes' to the institute; becoming influential figures as their ideas settled by finding traction.

I suppose methodologically, I'm advocating a kind of 'sub specie aeternitas' stance towards philosophy; look at it both anthropologically and materially, how do its concepts tend to develop. Maybe I should call it 'philosophical naturalism' just to be incredibly perverse. So when I'm making statements like:

'Problems have a habit of dissolving others in their posing'

I think I'm providing some description on the former level. This is how it tends to play out, without any pretensions of logical necessity. But it seems like there's a conceptual generality in how it plays out; and I'm trying to argue* that this problemscape view is a true analogy between philosophy and other types of inquiry.

*: what I actually do is express observations in a series that have, to my mind, shared content and insert words like 'thus' and 'therefore' between the sentences or sufficiently distinct ideas.

edit: I don't mean to suggest that all pioneers are left field mavericks, the distinction I draw between philosophy in a research program and philosophy following your own nose isn't that clear cut; but I think it communicates better than dealing with the inbetweens... ceteris paribus eh?

Oh absolutely not. What philosophers think and how philosophers philosophise, their treatment of other philosophers and what they care about - those propensities and expressive actions make philosophy what it is. That's part of what makes the delimitation of what philosophy is a philosophical problem; philosophers demonstrably do care a lot about what's required for philosophy; or more gently what makes good or interesting philosophy. But all of this is motivated by ideas of relevance and what problems motivate the philosophers; they get their implicit definitions from such prefiguring activities as questioning questions, formulating new ones, synthesising old ones, inventing modes of reasoning as they go, inventing problems and reinterpreting philosophical history in that light.

Fundamentally, this is because they made their concepts sensitive to other things (problems!); like force was in Newton's natural philosophy; a conceptual machine to answer problems about motion. As an aside, something that I find interesting here is that Kripkenstein functions a lot like a philosopher in philosophical discourse despite being an interpretation of one by another. The force laws function a lot like Kripkenstein in classical mechanics; they're a conceptual device which lets you address a lot of problems since the concepts are tailored to their problematic.

Any definition of philosophy would be contained in a problematic of this sort; and is likely to be a representation of the philosopher's problems of interest as well as their personality (or 'conceptual persona' like Kripkenstein). Laruelle attempts a definition of philosophy; philosophers inventively incorporated it and found (and were motivated by) new avenues to channel thoughts down.

Wittgenstein in the Tractatus casts most philosophy as a shadow in the sense you have outlined though; certainly a way it can operate. Basically what I'm saying is that the lack of an all encompassing good definition of philosophy is itself contextualised within the problems of philosophy; and while I'm certain that an absence of the definition can act as a motivation or problematic itself (like what you're doing), most philosophy doesn't seem to proceed like that and so can't take this definition as part of its 'native' nature.

Which dovetails nicely with what @StreetlightX is saying, but I do have a subversion:

Explanation occurs in medias res, and not sub specie aeternitatis.

really interesting stuff happens when people try to assume that perspective. I imagine Deleuze does this with 'absolute de-territorial-ization' and 'planes of immanence'; just like when the most recent Abel Prize winner adopted that kind of synthetic, highly abstract problem space to expose links between number theory, geometry and harmonic analysis (not that I understand any of the Langlands program).

Oh, also; I discovered a food allergy last night, got one hour's sleep and I'm running out of pile cream. I'm probably not in the best place for understanding detailed prose at the minute.

I'm not going to assume the silly Socratic game of asserting power by questioning you on what you mean by terms, I'm responding in the way that I am precisely in order to try and convince you that it really doesn't matter; that is, it has no meaning for philosophy, to have a sufficient and necessary condition for what it is. Or even a more lax dictionary definition (they exist, of course).

Google/Webster gives it as:

the study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline.

I'm reading you as equating lack of definiteness in specification and lack of any substantive characteristics. This is suggested in:

It becomes a problem for me when the ambiguity about definition is used to shut down lines of enquiry others are finding useful. Too often I hear "Logical Positivism has been disproven", "Kant showed that...", "[X, y or z] is not even proper philosophy", "you can't comment on X until you've read y".None of this has any justification without a definition.

Philosophy is, by necessity, everything that isn't something else.

I'm pretty sure that those shut down attempts are part of the positive character of philosophy. Like what distinguishes it from rhetoric; philosophers are supposed to care about the true nature of things. A speculative realist and a correlationist are going to be at odds methodologically; some threads of ideas will see problems in other threads of ideas which aren't native to that thread of ideas... And characterising what is and isn't native (characteristic, necessary for, required by, condition for the possibility of, presupposing the material conditions of...) to a set of ideas often in novel ways is absolutely part of the analytic stock and trade of philosophy.

J.L. Austin:

Ordinary language is not the last word: in principle it can everywhere be supplemented and improved upon and superseded. Only remember, it is the first word.

Heidegger:

Not only that. On the basis of the Greeks' initial contributions towards
an Interpretation of Being, a dogma has been developed which not only
declares the question about the meaning of Being to be superfluous, but
sanctions its complete neglect. It is said that 'Being' is the most universal
and the emptiest of concepts. As such it resists every attempt at definition.
Nor does this most universal and hence indefinable concept require any
definition, for everyone uses it constantly and already understands what
he means by it. In this way, that which tP'- ancient philosophers found
continually disturbing as something obscure and hidden has taken on a
clarity and self-evidence such that if anyone continues to ask about it he
is charged with an error of method.

and it is a thoroughly excellent part.

I'm not interested in defining philosophy when the definition of philosophy would itself be a philosophical problem. Inquiry proceeds without a satisfactory definition of it, so do all of its manifestations. I'd be as well wondering how you could post without being able to give a necessary and sufficient condition for a given object to be 'part of a language'.

Even math has somewhat ambiguous boundaries - at what point does it become logic? Is set theory logic, like it is historically, or is it part of mathematics? Is it as some mathematicians treat it like 'metamathematics' - then what about model theory, is that mathematics despite sometimes being called metalogic? Where does pure mathematics end and applied mathematics begin? When does statistical physics collapse into statistics or applied statistics become statistical physics?

Those questions need answers, the continuation of every single discipline requires immediate clarification of what they're doing!

I think all of these are largely pointless questions. If you insist that something have a clear definition before we can begin talking about it, or even begin to circumscribe its sense (how are definitions made without this stage of prefiguration?), I've got no interest in continuing the discussion.

That the features we're talking about are also features of inquiry in general is largely the premise of this thread.

I mistook your intentions as a usual 'what's the point in philosophy anyway' poster, albeit an articulate one.

I think you're right that it's a feature of inquiry. But it's also a feature of worldview, perspective and personality. I'm sure you'll have had arguments with people, say housemates or partners, where one of you did something completely usual and inconsequential for you but to them it's a big deal - a gripe or a blessing. The same can be said for problems in philosophy; though, problems in philosophy motivate more philosophy; some of which is developing new problematics. Which is probably related to how the philosopher (and philosophy) looks out on the world and what problems they're motivated by. Sometimes it's intimately personal and political like Bell Hooks, sometimes it's abstract generality like Kant.

Philosophy seems pinned between two attacks, those who demand it reveal what 'truths' it has discovered or else be consigned to the rubbish heap, and those who claim the whole thing is nothing more than a series of works of art, you either like or you don't.

I think you gave the conditions under which this is a non-problem in your post, ironically. All inquiry functions like that, thinking critically, rationally and creatively are all part of inquiry in general - they don't suddenly disappear or lose their general character when the inquiry is philosophical in nature.

The thing to note here is that it's pretty rare that a philosophical problem could be called 'purely philosophical'; philosophers typically care about a definite space of problems. Witty cared about doing right by language, Heidegger cared about doing right by being and subjectivity, the positivists cared about grounding science in a scientifically flavoured philosophy; doing right by science. So did Lakatos, Kuhn and Popper in related but contrary ways. Foucalt held science in something like an anthropological epoche and cared about it as a discursive practice. They all cared about something, and they inquired in the way people inquire; inventing along the way and following their noses along historically conditioned trails of expressions (islands of sense in the previous post) to do it.

I also read you as dismissing what I had to say because the vocabulary in my responses to @csalisbury was unusual. We usually have a similar perspective on things and similar philosophical tastes, so I can use a playful shorthand with him.