Not to mention the fact that it allows Derrida and Russell to be considered as part of the same subject.

Depends who you ask - I know the Cambridge faculty of philosophy (at least at one time) would have rejected any claim to the effect that Derrida was a philosopher. Russell, of course, was truly venerated as one. But then, that's probably grist to your mill :wink:

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).

A particular line of reasoning can have internal consistency, but there are so many theories from so many avenues, that can aim at solving a certain question... — schopenhauer1

I'll stop you here; again, you're changing the language: it's not 'solving questions' at stake: it is posing problems, determining the concepts through which problems themselves will be posed. What 'Wittgensteinians' or 'Schopenhauerians' or [etc] tend to agree on is not a 'solution', but the way in which a problem is posed; they differ, on the other hand, in what they draw attention to, in what they consider significant or remarkable. It should also be mentioned that this happens in math more than I think you're willing to let on: the fact that, say, ZFC axiomatics underlies mainstream set-theory is anything but a natural 'given': there are plenty who contest it, on pragmatic grounds. Or else consider the occasional question of whether set-theory or category theory is the most appropriate 'foundation' of math; B&C themselves contrast different notions of infinity, with neither one 'naturally' better than the other.

These are all things that 'catch-on' on the basis of pragmatics; they're all 'machines' that are well-tailored to working with certain inputs, and not others. Tools, liable to be put down in favour of other, different tools if necessary. Philosophy is a tool-kit, just like that.

Or put otherwise: there is no 'ultimate symmetry', the breaking of which explains individuation; it only seems that way after-the-fact, once you've illegitimately abstracted the concept from the conditions which gave rise to it; Symmetry is always-already broken in some way: there are generalities and particulars, and even stratified hierarchies of such divisions, but they develop from the 'bottom-up', even if, once so developed, the higher levels attain a consistency of their own (e.g. category theory as a 'response' to problems in algebraic topology). Explanation occurs in medias res, and not sub specie aeternitatis. — StreetlightX

This is the important first choice of applying mathematics, right here, the defining of "individuation". If we assume that the multitude, the many, or all, are one (the universe), as a bounded object, then we must assume that this bounded object has a value, a finite number, or else the multitude, the many, produced by division is unreal. Any division which allows for infinite division of a bounded object is a completely arbitrary division. This form of individuation denies the possibility that the many, in thje sense of the real, true many, is infinite, and the possibility of infinite numbers, in any real sense, is excluded. But if we define individuation such that we allow that the individual is a real object amongst the many, then the multitude is real, with the real possibility of infinity. However, then we need support for the assumption that the individual is truly the individual, in the sense of indivisible.

In reality we must accept the two distinct choices, each as valid choices, each with incompatible ontological implications. The incompatibility has to do with how we define "one", and is inherent within the concept of numbers. Does "one" signify an indivisible unit, or does it signify a divisible unit? Numbers like 2, 3, 4, represent divisible units, 2 representing a unity which is divisible into two distinct units. But 1 when understood in this way must be indivisible. If we allow that 1 is divisible, we undermine the meaning of unity. But we need to allow that one is both a unity and is divisible, so we allow two incompatible, contradictory concepts to coexist within one, being signified within one symbol. The number 2 for example, signifies a unity (one), which cannot be divided without negating the unity which is signified by the numeral. However, at the same time, it signifies two distinct unities. So each number signifies a type of divisibility and a different type of indivisibility, both at the same time.

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. — fdrake

It's funny you should mention Kripkenstein, I've often used him (presumably a him?) as an example of the difference between the framing of a philosophical proposition by the author prior to publication, and the framing by the reader afterwards, a bit like literary deconstruction (as opposed to philosophical deconstruction with its historical baggage). I like the fact that Kripke interrogates what Wittgenstein means to him, not what Wittgenstein means sensu lato.

I'm very much in favour of more critical analysis of this sort, but things, if anything, seem to be swinging the opposite way in favour of a more historical exegesis. Except in places like this.

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. — fdrake

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.

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?

Philosophies of the event (as a sociological phenomenon) correspond exactly to the inability to unironically and sincerely hold some kind of value (related to action, not thought) that can be actually acted upon to produce an 'event'. — csalisbury

I think you're just... wrong about this. I mean, yeah, the question of values is something so far underdeveloped in this thread, but the emphasis on pragmatism is conceptually inseparable from acknowledgement of the role that values must play. I mean, I think (maybe??) you're getting the wrong idea from the vocabulary of 'choice' which yeah, rings with all kind of 'voluntarist' associations. But analysing it this way - and it's pretty formalist, I admit - doesn't (yet) say anything about the conditions under which such 'choices' must be made. And nothing I've said precludes the idea that "choices/decisions involve the whole heft of your spiritual being" - which I think is entirely right!

At this point I don't even know if we agree or disagree with things. You're being much too meta for me, I can't keep up, well done, you're winning the prize?

These are all things that 'catch-on' on the basis of pragmatics; they're all 'machines' that are well-tailored to working with certain inputs, and not others. Tools, liable to be put down in favour of other, different tools if necessary. Philosophy is a tool-kit, just like that. — StreetlightX

But again, this "catching-on" in mathematics, eventually moves to consensus. Thus, even debates over axioms about infinities, etc. will eventually get to a point via demonstrable proofs that convince the community that this should be included in standard views of the problem, until someone else brings up an issue. 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. This isn't a matter of degree but completely different starting points. It's like one is using axioms and the other is using poetry. How is that commensurable for a consensus and step-wise branching out that occurs in the math world?

But again, this "catching-on" in mathematics, eventually moves to consensus. Thus, even debates over axioms about infinities, etc. will eventually get to a point via demonstrable proofs that convince the community that this should be included in standard views of the problem, until someone else brings up an issue. — schopenhauer1

But, to be blunt - this is wrong. I'm not denying the fact of consensus - clearly Cantorian infinity and ZFC axiomatics generally win the day - but they're accepted on the basis of their usefulness... right up until they're not; and the point is that this doesn't differ in kind from philosophy. I mean, take differential calculus. What's the best approach? Geometric? (qua Netwon?) Arithmetic? (Delta-Epsilon?) Non-Standard? (Leibnizian flavoured)? But there is no 'best' approach because 'best' is only ever relative to what you're trying to do with the calculations. Again, I think you're far overstating the kind of consensus that actually exists in math.

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]

But, to be blunt - this is wrong. — StreetlightX

What's the best approach? Geometric? (qua Netwon?) Arithmetic? (Delta-Epsilon?) Non-Standard? (Leibnizian flavoured)? But there is no 'best' approach because 'best' is only ever relative to what you're trying to do with the calculations. — StreetlightX

I'm going to be more charitable. You are both right and wrong :p. You are right; there are different ways to solve problems based on what they are trying to do with the calculations. Do they work based on the problems they are trying to solve? Yep. Can it be demonstrated? Yep. You are wrong in that if it is demonstrable, then it can live alongside the other demonstrable methods. It allows for diversity, but this diversity, even if a method is not liked because of its angle of attack on the problem,cannot be denied to have solved the problem. If another angle is seen to solve this, that, AND the other problem, then slowly a consensus will build on that new method. However, the claim still remains- demonstrable results and consensus make mathematical creativity/novelty different than philosophical creativity.

Stop. Talking. About. Solving. Problems. Start. Talking. About. Conceptual. Determination.

How do you know if category or set theory is the best approach? How do we know if or how the math can capture the freak waves?

Yes, I'm leaving it open-ended for reason.. Socratic method and all..

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.

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

But why are Cauchy cuts and Dedekind Cuts both considered valid moves?

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

Good answer. But let me take a different approach. Why would a poem or a piece of music or even a mathematical proof that models a different problem not be seen as a valid move? I’m not being cheeky here. I am going somewhere..

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.

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. — fdrake

Ok, but I think you would say it better than me probably. But my sentiment I'm trying to convey here is that the models and demonstrations you speak of have a lot of constraints as to what kind of methodology can be employed to solve a particular mathematical problem. Philosophy does not have these constraints (unless your philosophy is to put certain constraints on, but then the argument about what constraints to put on would still be contested and so on). There is a certain consensus in the math community about what counts as even in the realm of what is valid for an answer. From there, further proofs and demonstrations would be needed to justify the weight of one modeling claim over the other. Eventually, if it happens enough times in enough places, another consensus takes place (this time, not on the constraints of what a valid answer can look like, but on what is the better model). Examples of this being the use of irrationals, non-Euclidean geometry to solve problems of relativity, etc. etc.

Edit: Sure, a piece of music as an answer in philosophy would also be considered by most as outside the realm of philosophy, but the scope is very wide, to the point of it being negligible to speak of an "agreed upon" constraint in the philosophy world.

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. :)

Haha, I hope to continue this at another point. Thank you for hearing this through.

What's interesting to me about the discussions here (the reason I enjoy reading them for the most part) is the fact there is an attempt to discuss topics without prescribing the framework in any way whatsoever. I'm fairly certain this never works, but the ways in which it fails are what fascinate me. Doing research, one never encounters this, but here, it's the most common approach, even if a philosophical school is specified.

This probably requires a bounded terrain; if everything can be relevantly said of an idea it says nothing. — fdrake

This is part of what I'm getting at, phrased this way it begs the question. Surely we should first ask if everything can be relevantly said of an idea and so render it mute. There's nothing wrong with subsequently framing it so as to constrain what can be meaningfully said of it, but we need to know why we're doing that. Not in academic research (there the bounding is done for you), but on a forum like this it's essential, and I certainly feel like there is little acknowledgement of that (maybe little understanding, but I wouldn't like to judge).

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. — fdrake

I don't win myself any friends by my habit of picking at propositions until they fall apart, but having missed that ship long ago, I'll have a go at this one. 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?

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. — fdrake

I like this. It's what I'd like to think I'm doing here too (if that's not too presumptious) it's just that I'm considerably less charitable than you in my interpretation. I think there's a lot less influence from internal drivers (the structural constraints) and a lot more social psychology in describing how concepts tend to develop.

Hey Pseudoynm, thanks for bringing up that earlier reference to fdrake where he said this:

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. — fdrake

I can see now, that fdrake was getting at a similar notion I am in terms of difference of philosophy and math. So I was just getting to your own point fdrake :D. Too many frameworks with radically different methodologies attacking similar problems. In math, the constraints of community are really just constraints on what math describes (numbers, relations, patterns, measurement etc.). Philosophy is way more open-ended and thus the consensus is little, quantifiable information can be manipulated in constrained ways (you can only do so much with quantifiable information) contra philosophical problems, and the kind of answers are open-ended in philosophy contra math. But this is because the very nature of philosophy is how unconstrained it tends to be.

However, the claim still remains- demonstrable results and consensus make mathematical creativity/novelty different than philosophical creativity. — schopenhauer1

I don't think complete consensus is ever possible in mathematics until the complete nature of reality is completely understood by everyone. Then everyone will agree on which mathematical principles ought to be applied to which aspects of reality. Until then, we will each have our own metaphysics, our own ontologies, and apply mathematics as suited to these various ontologies.

Since consensus on mathematical principles is dependent on consensus of ontology, it is impossible that there could be a higher degree of consensus on mathematical principles than there is on philosophical principles.

Since consensus on mathematical principles is dependent on consensus of ontology, it is impossible that there could be a higher degree of consensus on mathematical principles than there is on philosophical principles. — Metaphysician Undercover

All that needs to occur is that a higher amount of constraints that needs to take place math than in philosophy. I'll even confine it to just an area like metaphysics/epistemology. In these realms of philosophy, the constraints are so wide that there is no consensus to justify which move is more valid than another. In math, there are at least some moves that are universally considered invalid. I could not solve a mathematical problem with a treatise on "being" for example. However, a metaphysical argument might be framed as a problem of "being", a problems of propositions/linguistics, problems of a priori synthetic knowledge, problems of empirical data gathering, etc. etc. It is framed too broadly for even a consensus on what a valid answer looks like (unless you fall within a camp with another philosopher who shares that point of view, but that doesn't negate that philosophy itself is much broader outside this compartmentalization). Thus I said earlier:

But my sentiment I'm trying to convey here is that the models and demonstrations you speak of have a lot of constraints as to what kind of methodology can be employed to solve a particular mathematical problem. Philosophy does not have these constraints (unless your philosophy is to put certain constraints on, but then the argument about what constraints to put on would still be contested and so on). There is a certain consensus in the math community about what counts as even in the realm of what is valid for an answer. From there, further proofs and demonstrations would be needed to justify the weight of one modeling claim over the other. Eventually, if it happens enough times in enough places, another consensus takes place (this time, not on the constraints of what a valid answer can look like, but on what is the better model). Examples of this being the use of irrationals, non-Euclidean geometry to solve problems of relativity, etc. etc.

Edit: Sure, a piece of music as an answer in philosophy would also be considered by most as outside the realm of philosophy, but the scope is very wide, to the point of it being negligible to speak of an "agreed upon" constraint in the philosophy world.

I also said:
I can see now, that fdrake was getting at a similar notion I am in terms of difference of philosophy and math. So I was just getting to your own point fdrake :D. Too many frameworks with radically different methodologies attacking similar problems. In math, the constraints of community are really just constraints on what math describes (numbers, relations, patterns, measurement etc.). Philosophy is way more open-ended and thus the consensus is little, quantifiable information can be manipulated in constrained ways (you can only do so much with quantifiable information) contra philosophical problems, and the kind of answers are open-ended in philosophy contra math. But this is because the very nature of philosophy is how unconstrained it tends to be.

Does "one" signify an indivisible unit, or does it signify a divisible unit? Numbers like 2, 3, 4, represent divisible units, 2 representing a unity which is divisible into two distinct units. But 1 when understood in this way must be indivisible. If we allow that 1 is divisible, we undermine the meaning of unity. But we need to allow that one is both a unity and is divisible, so we allow two incompatible, contradictory concepts to coexist within one, being signified within one symbol. — Metaphysician Undercover

Remember that in maths, a unit is defined by the identity element - a local symmetry that can't be broken by whatever operation broke the global symmetry. So 1x1=1. Or A-0=A. The fundamental unit is whatever emerges as the local limit on symmetry breaking. The act of quantification results in a quantity where the action no longer makes a difference. Things finally stop changing. You arrive at a fundamental grain so far as that symmetry-breaking is concerned. Now nature just spins on the spot, quantified in good atomistic fashion.

This is indeed the tale of fundamental particle physics. So maths and physics are talking about the same universal mechanism. Reality exists because there was a symmetry to be broken. And then the breaking of a symmetry eventually also hits some local limit. A new state of symmetry is discovered where the individuating, the differencing, no longer results in a difference. You wind up with a smallest Planckian grain of action.

So geometry begins with the fundamental thing of a zero-d point. Dimensionality cannot be constrained any more rigorously than a dot, a minimal dimensional mark. Having found the stable atom, the concrete unit, the construction of dimensional geometry can begin.

Instead of a holistic metaphysics of constraint - the story of how a unit or identity element could naturally exist at the end of a trail of symmetry breaking - we can flip to the more familiar reductionist task of (re)constructing the world from the bottom-up. We have our unit. We can then start framing the universal laws that then do arithmetic with that unit, building a reality up step by concrete step in accordance with a material/effective cause notion of how the world "really is".

So in the mathematical realm where 1 is the identity element - the unit that is unchanged by the kind of change that more generally prevails - it is both part of that world and separate from it. It has that incompatibility which you point out. And that is because it is a re-emerging symmetry.

Globally, a symmetry got broke by the very notion of a division algebra. Division, as an operation, could fracture the unity of the global unity that is our generalised idea of a continuous wholeness - some undifferentiated potential. But then divisibility itself gets halted by reaching a local limit. Eventually it winds up spinning on the spot, changing nothing. A second limiting state of symmetry emerges ... when our original notion of unity as a continuous wholeness finally meets its dichotomous "other" in the form of an utterly broken discreteness.

So it is the usual metaphysical deal. A dichotomy that finds its fullest resolved expression in the form of a local~global hierarchy of constraint. To bound a world takes opposing poles of being. And this is what both physics and maths have worked out - even if the holism that underpins the successes of reductionism is not itself generally appreciated.

It does seem weird. Even science and maths don't really understand why they work so well - why they get at the basic structure of existence. Everyone thinks it is because of their reductionism. And that is certainly what works in a "pragmatic" everyday sense - when the mechanical and atomistic view is good enough to serve our very concrete human purposes.

But that is why Peircean pragmatism, the original metaphysical kind, is important. Existence is a story of how constraints can tame flux or instability, eventually resulting in the irreducible grain - the fundamental units or atomistic actions - from which a resulting counter-action of mechanical constructability can start.

Bottom-up material/efficient causation can be a thing once top-down constraint has forced everything towards a local limit and a symmetry has emerged there which can be the foundation for more semiotically complex constructions.

So I think your whole approach mistakes description for prescription, effect for cause: once you suck the life out of problems-in-duration and make the move into a higher dimension where everything can be seen from the perceptive of placing them into neatly-parsed boxes (accidents or necessities? generalities or particulars?), then and only then does development seem to proceed on that basis; but the leap into that dimension is illegitimate: it's simply retroactive ratiocination, the work of philosophical morticians. — StreetlightX

Is there any real effort at thought behind these ad homs?

Sure, there is pragmatism in the weak Jamesian sense of utilitarianism - whatever is good for "someone's" contingent purposes.

But the existence of the ad hoc itself highlights the "other" which is Peircean pragmatism - the metaphysically general kind. Instead of the someone, we are now talking about the generality that is "anyone".

So you can wave the flag for the ad hoc story. It is part of my larger story already. It is precisely the kind of contingency that I am generalising away as the differences that don't make a difference when the intent is to reveal the basic structural mechanism at the heart of existence.

Or put otherwise: there is no 'ultimate symmetry', the breaking of which explains individuation; it only seems that way after-the-fact, once you've illegitimately abstracted the concept from the conditions which gave rise to it; Symmetry is always-already broken in some way: there are generalities and particulars, and even stratified hierarchies of such divisions - all this can be granted - but they develop from the 'bottom-up', even if, once so developed, the higher levels attain a consistency of their own (e.g. category theory as a 'response' to problems in algebraic topology). Explanation occurs in medias res, and not sub specie aeternitatis. — StreetlightX

I've explained this to MU above. And I already pointed out that the Peircean view is ultimately triadic - an orthogonal pair of dichotomies - in that it combines a diachronic view with the synchronic. It is a tale of two symmetry-breakings - the developmental one that goes from vague potential to crisply ordered, and the scale hierarchy one which is the developed crisp outcome, the equilibrium state where the local constructing is in stable balance with the global constraining.

So there is room enough in the triadic view to house all your metaphysical concerns. :)

There is bottom-up atomistic construction, for sure. There is actualised contingency or degrees of freedom, for sure. All the regular stuff that makes reality safe for reductionist thinkers can be found in a world that has developed enough to have attained a stable local~global structure.

Once the world is being held in place by the constraints of its own history, it does look securely classical to the typical human observer, sitting right at the Copernican centre of the story.

And you can join all those who blithely takes this as Humanity's right - to see themselves as the spiritual centre around which the Universe then dumbly and mechanically revolves. It is our own personal desires and values that matter - which should inform material reality. We impose ourselves on nature in some metaphysically rightful fashion as the world itself is just some bunch of ad hoc events, lacking any formal or final causes. What completes existence is us, at the centre, doing our thing of expressing ourselves in some kind of glorious free pluralistic fashion.

Talk about reinventing theology.

Meanwhile, science and maths are just getting on with the job of revealing the deep structure of existence. And the reason why they are working can only be understood once you see how they are simply an expression of a Aristotelian/Peircean holistic structuralism.

Folk are confused because the first fruits of maths and science were the presentation of a classical world - the world of the atomistic, mechanical, local, deterministic, etc. That exploded like a bomb in people's thoughts.

But now it is clear how the classical realm is emergent. It is what you get only as existence develops its generalised habits of constraints and forms a clear local~global structure as constraint eventually produces the grainy local limit where atomistic construction actually starts to be a thing.

So the new project is holism. And mathematical physics is deeply engaged with that. It seems to know what it is doing at least.