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.

Absolutely, I agree, but that's rather the point I'm trying to make (I'm guessing you're either no Wittgenstein fan or you have a completely different take on his project to me). What was Wittgenstein's conclusion of this critique?... Philosophy is only descriptive, its purpose therapeutic. The only problem to be solved is that of the human psychology.

Heh, completely different take it is. I'm invoking him in support of my position, not against it! Well, perhaps not completely different. I think Witty drew the wrong conclusion from his own argument, which ought to point to what it means to formulate good, well articulated philosophical questions - not reject philosophy altogether. He didn't recognize that philosophy - any worthy of the name - already operates according to the strictures he rightly understood language to abide by. Which is what happens when one has as poor a grasp on the history of philosophy as Witty, unfortunately.

I think Witty drew the wrong conclusion from his own argument, — StreetlightX

Well, you'd be in good company, some of my favourite philosophers are also of this opinion, but not me. I have a lot of sympathy for such a position, and I can just about see how you can reasonably make the link you're making here, but I just don't find it convincing enough... yet.

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. — fdrake

Absolutely, I think I understand what you're saying here, for maths. I'm afraid I'm still not quite seeing how it applies to philosophical problems. I mean, I can see exactly how it could apply, just little evidence (certainly in my limited experience) that is actually does.

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.

I'm quite content that inquiry in general is like a landscape, and the problem is like flowing water, it's route being almost dictated by the structures in the landscape which guide it in a particular, almost determinate, course. Its just that I think philosophy (particularly of the public kind) is not an inquiry of this sort at all, but more a description of what just happened, the process by which water flows through this landscape, a reassuring story (or various such stories). To me, problems in philosophy are almost always problems of translation. I'm trying to translate a description of the problemscape (if I can borrow you terminology) not run the water through it to see where it goes.

Yes. I mean I haven't mentioned him at all here but yes, 'how to step beyond Derrida' is massively written across all of this. Because yeah, it's actually a question that really messes me up, like, how do you move beyond the formalist promise of the à venir, of the 'mere' always-already opening to the future? (especially because I think it's entirely correct?). And I'm finding in this language of 'choice' precisely that way to think beyond Derrida's 'undecidables', those moments that both belong and do not belong to a system (like Godel statements...); But I'm also trying to think that move beyond in a very specific way, a way that isn't just a fall-back into a Russellian 'theory of types' where you simply avoid self-reference (even as you self-refer to do it), but in a way that affirms the productivity or the generativity of paradox, where this moment of two co-existing incommensurables force a leap of creativity to diffuse the tension. — StreetlightX

I guess my thought is that this is the limit of formal analysis (tho maybe laruelle... but I don’t know him except for some postcard synopses.) Or the limit of a universal formal analysis. I think you could bring the analysis to bear on actual moments (which tbf you are doing with the math stuff, tho—-) but in a way that the actual moments aren’t examples furnished in support of the model, but such that the model is more like a base camp, set up in order to better sketch a single moment in its haecceity (for lack of a better word.)

I think (and maybe this is laruellian?) that there’s no formalist (and philosophy is alway formal) way past the a venir. Once you’re there, then you just have to enter into it, which, to me, means bringing philosophy into uneasy commerce with something else. You mentioned ecology and so forth, but I have the queasy sense that any philosophical analysis of this sort is going to funnel into the insatiable maw of whatever it is that apo’s peddling. There’s still an anti-TOE TOE lurking, as apo likes to point out. But I don’t think this is just a rhetorical gotcha. It’s a real danger. I think apo’s wrong that there’s only two choices (Kierkegaard and Hegel) but I think he is pointing out something.

I really like the image, in Dante, of Virgil not being able to pass into purgatory. Not that reason is hell (tho maybe reason left to its own telos is?) but the idea that Dante has absorbed what he needed. He doesn’t leave it behind, but he steps over a boundary that reason alone would leave him approaching, retreating in endless oscillation. So like: an image of philosophy as a “class” in a rpg. Or a character in a poem or story: say one samurai of the seven. Training and mastery is important, to stand on your own with your own unique skill set but then: Something that is used ina broader conversation, but in a way that refuses to reintegrate the entirety of the conversation into itself. The best part of these kinds of conversations is that a meeting of philosophy and something other than it creates its own thing, the same way two people talking (as equals) are in a novel space which is irreducible to either of the two.

I feel like samz[]dat is an example of this. Nick Land is too, but an example of when it goes wrong. Proust, imo, is an example of when it goes very right. Of course the state of the art then was Bergson, so things have changed, but the process of going to the limits of the art, then setting it up to talk with other currents. Sloterdijk in my opinion is the best thing on the market right now. He’s not perfect and has his flaws, but If I had to cite someone I think is doing it right, it’s him. Spheres is the best work of ‘philosophy’ in the past fifty years. His training is Heideggerean, (tho he has facility with Derrida, Deleuze, Brandom etc) but he writes with his whole being and you get something legitimately new ( not merely novel)

Or like series in deleuze where a bunch of disparate ideas resonate, not because they’re all thought under the same magisterial umbrella, but because, taken together, they birth something new.

Tldr: to paraphrase a close friend “he’s identified the problem and knows that things have to change, but all he does is talk about it. He tells me how unbearable things are for him, and talks about what he needs to do next. But it’s been like that for years. I listened for a long time, but now, I just can’t do it anymore. I’ve been supportive for a long time, but I have to get on with my own shit.”

I think that works perfectly as a description of left preoccupation with the a venir. (Cf zizek’s signs of the future). It’s not wrong, but it’s a check we keep being told we need to wait to cash. Eventually you suspect this particular kind of check has wait-to-cash baked into it. Like those joke signs that say “free beer tomorrow.”

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.

Don't you have to go to a higher dimension to match curves to functions?

Yeah, you could call that the 'history' of the problem, I could argue that I would see that as the history of maths, not the history of that problem, which I think of in a more ontological sense. But then we're just getting into defining terms rather than saying anything interesting, and despite the inexplicable preoccupation with that kind of bullshit in philosophy, that sort of investigation bores me. I'll see if I can restate it in language that might prove less equivocal.

I don't think I would need to know why that equation needs solving in order to find a good solution to it. I only really need the language it's written in, some axiomatic presumptions which restrict solutions, and some signposts pointing in the right direction.

Philosophical problems share these features, but this is vacuously true. All that's being said is that problems all have at least some factors influencing the possible solutions.

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.

Maths may well have choices about which problemscape to use and the solutions will certainly be guided by that choice, but that's nothing like philosophy, where the problemscapes are carved, and mass-engineered with the sole intention of producing a pre-determined type of solution.

I think (and maybe this is laruellian?) that there’s no formalist (and philosophy is alway formal) way past the a venir. Once you’re there, then you just have to enter into it, which, to me, means bringing philosophy into uneasy commerce with something else — csalisbury

Yeah, look, I agree with this in its entirety (it's Deleuzian becoming! Becoming-friend; Becoming-'The-something-else'...), but fuck, man, it's not easy to do. But - to pick up some recent themes - this 'hardness' isn't just a psychological quirk ("we're not built for it" or whatever), but an issue endemic to reason as such (transcendental illusion, etc). I mean - reason generalizes, that's just what it does (the machinery of token and type, etc). And argument and discussion presupposes this machinery as a matter of course ('justify your position'; 'defend it against it's negative'; 'how does it deal with this case?'), so it's incredibly easy to get caught in it (and if one doesn't 'get it' the previous sentence sounds like madness).

So you literally have to reason-against-reason or employ a kind of trans-reason that works diagonally across the tiered distribution of token-type stratification that comes so naturally to us (this is why Deleuze was so against the negative, and always said that philosophers 'run away from discussions': it takes away from the positivity of concept-creation that responds to the encounter, the becoming-X that is it's result). So I'm not saying this (just) to excuse it but I definitely get caught up in the game of 'giving and asking for reasons' and all the attendant reflexive loops which lead one higher and higher up the reason-ladder until you reach the level of formal purity which yeah, I often circle around. Nothing particularly wrong with this.

So the best way to disable the TOE is simply to... ignore it. Or at the very least you point out why it's nothing but a self-sustaining circle, and leave it to its own devices. You create instead. It's true I guess that there's no real way to say 'you are wrong because...', and create off the back of that, because it immediately commits you - structurally, as it were - to being gobbled up by the Absolute Cricle of Circles (idealism of 'belly turned mind'), but goddammit you can't just not do it either, at times, as much as it's nice to be above the fray. Getting messy is fun. But yes, yes, more becoming-other (too philosophical?), I get it.

Thinking about this more... Systematization is like The Game. The Game is unwinnable and only has one rule: don't think about the Game. You lose when you think about The Game. You've just lost by reading this (traditionally, you get a 30 minute 'cool-off' period before you can lose again). System is the same. And Derrida's whole shtick about undecidables is basically this: the undeciable means you can always be inscribed in a system if you look for it (read: 'decide to be so inscribed'). It's the fridge light of philosophy: if you look for it, it will be there. It's empty, brutal tautology. So you don't look for it. You let thought be forced by the pressure of the encounter-motive (becoming-ecological, becoming-math) and you just create. But as soon as you separate that creation from 'what it can do' - its pragmatics - and think about it in terms of system, you lose the Game, even if just to argue that it is not a System.

So at least one lesson of Derrida is: keep moving. Don't let yourself get pinned down. And with Deleuze there's no imperative to movement - he kinda just... does it. There's no concern for the spectre of Hegel, no real engagement - other than bald-faced denunciation; he speaks of his 'innocence' and 'naivety': what Nietzsche tried - and ultimately failed - to do. Having said all this, I've got 30 minutes before I become undecidable again.

Anyway. Literally shower thoughts.

Alsooo, there seem to be some here who think that the kind of 'math problem' I'm comparing philosophy to is some stupid banality like "2+2=?". But this is not a genuine problem of or for math ('of' understood in the subjective genitive case), there is no concept to be determined here. "4" clearly 'falls out' of it, and the only 'problem' is how to go through the motions of calculation. The kind of math problem I'm referring to is of the kind detailed in the OP: What to do with the irrational? Or: What to do with the infinite? Or: What to do with the imaginaries? Not some trivial kindergarten crap where you simply map domain to codomain.

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'.

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. — fdrake

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.

What proceeds from this is the point I take issue with, the idea that there can still be 'wrong' solutions given agreement of certain hinge propositions, that there can be objectively 'uninteresting', 'non-useful' or 'unnecessary' sets of hinge propositions, or that some discourse (no matter how so constrained) can somehow still be 'measured'. Measured against what scale I must remain unenlightened, as SLX seems to have closed that enquiry.

Maths seems to me to fit all these criteria well (proofs from within agreed axioms are measured by the same metric, problems are agreed on as being problems). Philosophy quite evidently does not. There is no agreement on solutions even within accepted hinges, it remains entirely undecided what problems are necessary, interesting or useful after 200 years of inquiry. The problem, using hinge terminology (though I wouldn't personally use the term), is that hinges in philosophy are not a single early choice like they are in maths, they are a continual hierarchy of choices which is permanently in flux at the terminal branches. One selection does not produces a necessary solution, it produces another choice. That decision just yields a third set of choices, that one a fourth and so on, we are forever choosing hinges, never having the resulting discourse.

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.

So, philosophers are full of shit when they talk about hinge propositions?

The only hinge proposition that I've encountered throughout my time doing philosophy is solipsism. What do you chaps think about that hinge proposition?

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. — fdrake

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.

Each of us is using a different framing device. This is supposed to be an impossibility, but it's not. — fdrake

How have you concluded that it's supposed to be an impossibility? I can't see how this is implied.

To operate on this level of abstraction has as a hinge that we can take other hinges and philosophise about them. — fdrake

Absolutely. So is such a hinge useful, in that it allows us a discourse we find useful? Certainly, if philosophy does anything at all, then allowing a discourse about its aims and methods should also provide some utility.

But we're straying from the point of the thread (I think). We got here by way of my claim that philosophy remains significantly different from maths, and what similarity it shares is of trivial importance. To maintain that claim, in the terminology of hinge propositions, it only need be the case that anything equivalent to a hinge proposition in maths (which I take to be something like axioms or accepted methodology?) dictate solutions to the problems within that frame in a way that does not happen in philosophy. I understand that in higher level mathematics (not my "kindergarten crap"), there will be debates about what constitutes a 'solution', but I'm afraid without a raft of evidence to the contrary (and none such has been advanced), I remain of the opinion that the levels of disagreement about what constitutes a 'solution' in maths (as a whole) are dwarfed by the wholesale and almost exhaustive disagreement on the same question in philosophy. An explanation for that is the key question for meta-philosophy.

So, philosophers are full of shit when they talk about hinge propositions? — Posty McPostface

I'm not sure I understand your point here.

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.

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. — fdrake

By definition, that fulfills the criteria of solipsism. Don't you agree?

I'm not sure I understand your point here. — Pseudonym

It was just a banal statement, irrelevant to the point of this thread really. My point is that if certain fundamental truths cannot be ascertained by philosophy, then what are your options about finding out about hinge propositions (or, I think, 'brute facts') if that's what the domain of science pertains to?

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,

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, — fdrake

Something we haven't talked about -- and I'd really like to hear your thoughts on -- comes out in the article @StreetlightX linked: one reason the proof was ignored is because it was a bad proof. That is, everyone expected a proof that would show what the connection to convex polygons is (if I'm remembering the issues correctly), but it turns out there's a purely statistical proof that leaves those connections, which have emerged in the years of attempted proofs, entirely unexplained. The reception seems now to be, well it's nice to know for certain that it's true, but that's not really what we wanted.

A proof being good or bad depends on what you wanted out of it. I could see this proof being eventually recognized as good if it leads to some deep insight about statistics. (There really shouldn't be proofs that are bad in an absolute sense.) Such a deep insight might even eventually link back to the algebraic geometry it passed by.

(Sorry if I'm garbling the math -- these aren't areas I know at all.)

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

Whenever @StreetlightX talks about creative concept construction, I always think about this good proof vs. bad proof thing. Good proofs are the ones that show you why the thing is true, and are crucial to the pedagogy of mathematics.

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.

“fridge light of philosophy” was great. Gonna respond when I get a moment

@fdrake @Srap Tasmaner:

My favourite example from the OP's paper was actually its discussion of infinity, which I didn't bring up for the sake of space - but since we're in the thick of it: it makes a comparison between the Cantorian notion of infinity - which is more or less accepted as standard today - and another, 'forgotten' attempt to think about infinity, drawn by John Wallis (the guy who 'invented' the number line and the infinity symbol, ∞). Wallis' argument was basically that infinity must be less than any negative number (contrapositively: all negative numbers are larger than infinity!), because:

1. As 1/x approaches 0, x = ∞
2. As 1/-x approaches 0, x also = ∞
3. But 1/-x > 1/0
4. So ∞ < -x (infinity is smaller than any negative number!); Which yields a new number line:

0 << 1 << 2 << 3 << ... << -3 << -2 << -1.

(See the paper for a more detailed, less condensed account). As B&C point out, this is mostly seen as a bizarre result and is mostly ignored today. But, as they note, we tend to have a very selective view of what is and is not 'bizarre': it's well known that Cantor's infinity gives rise to weird results too, in the form of 'Hilbert's Hotel' paradoxes, where, if we 'use' Cantorian infinity, we can show that there are as many perfect squares (X^2) as there are natural numbers (they can be put into a one-to-one correspondence), despite the fact that this too is incredibly counter-intuitive (the set of perfect squares 'ought' to be a subset of the the natural numbers).

So why do we accept one counter-intuitive result and not the other? Is there any intra-mathematical reason? But of course not. As Clark puts it in his solo book, "Which contradiction we allow to stand, it seems, determines which particular brand of infinite number we are inclined to buy". So what I wanna say is that this all plays into this idea that it's not the 'proofs' themselves which are decisive, but what it is that the proofs allow us to do; we accept and/or reject results not just on the basis of their intra-systemic consistency, but also on the basis of their fecundity, their fruitfulness for ... whatever it is we want to do (there's a mathematical empiricism here, that isn't at the same time a realism!).

Which brings us back to Witty...: "The mathematical proposition says: The road goes there. Why we should build a certain road isn't because the mathematics says that the road goes there - because the road isn't built until mathematics says it goes there. What determines it is partly practical considerations and partly analogies in the present system of mathematics."; (This also kind helped me think though some of the Zeno stuff that's been around the forum recently: all the questions re: calculus and the continuous and the discontinuous - the Zeno paradoxes probably arise from applying a certain concept to a phenomenon not suited for it).