I'm simplifying a little of course, but these commitments - exactly alike the commitments made in the determination of mathematical concepts - force upon a philosophy the kinds of contours it takes; To 'learn Hume' or to 'learn Plato' is not to learn simply what they said, but also why they said it, as well as the ways in which the distinctions they draw commit them - 'constrain them' - to saying certain things and not others. This is why there are - or rather can be - 'schools' of philosophy - as you said, 'Wittgensteinians', 'Platonists', etc; this would not be possible if not for that fact that philosophy sets it's boundaries - its distinctions, its categorizations - out in incredibly precise ways. As I said, to be a Wittgenstienian (for example) is not to 'say what Wittgenstein did'; it's to accept a manner in which problems are posed, problems which may be other than those even conceived of by Wittgenstein himself. — StreetlightX

Then you didn't see when I said:

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). — schopenhauer1

The point isn't that philosophy cannot be consistent and rigorous within its own framework, but rather that there are a plethora of ways to try to answer a question that come from radically different angles. Math is constrained by its very nature of describing quantifiable information (relations, measurement, numbers, etc. etc.). Thus the number of valid moves in math is much more constrained. Intra-mathematical methods can be contested (whether they are good), but limited because of the constrained nature of working with relations, quantities, and numbers rather than dealing with the foundations of reality and knowing itself. So there are different approaches, but they can only make so many valid moves in that world of constraints. This makes for a different kind of validity (mostly a more easily demonstrable one and consensus building one).

In philosophy, because of the open space of its canvas (so to say), even if an argument is rigorous and consistent, the fact that there are so many moves one can make to frame a problem (in other words, so many camps one can fall under) it really cannot be as analogous in its creativity as you are saying.

But I'm guessing from the hand-waiving that we've reached our usual limit to the extent you care to peruse these kinds of arguments and so further exposition would perhaps be pointless. — Pseudonym

On hand-waiving:

If someone said that he doubted the existence of his hands, kept looking at them from all sides, tried to make sure it wasn't 'all done by mirrors', etc., we should not be sure whether we ought to call this doubting.

On hand-waiving — csalisbury

Tell the guy looking at his hands not to worry, since you've relaxed the requirement for actually having hands in order to engage in hand-waving.

But again, do you want to claim that the connection is arbitrary? Do you have reason to believe that nature plays by different structural rules despite the evidence to the contrary?

To the first question, what connection are you asking me about being arbitrary? The connection between physical models and an independently existing nature? Well, I'd have to be a realist even to accept the terms of that question. Or the connection between physical models and the purported reality they claim to represent? This is not an arbitrary connection in the sense that the models are there precisely to model what the modellers take the models to be models of. The issue that got this whole post running is when models come up against a problem, what is to be done and are we free to choose arbitrarily what is to be done? The SM comes up against the issue that it doesn't account for gravity, and yet gravity is something that manifests in the physical world that SM purports to model. In come "gravitons" as a proposal to extend the SM to take into account gravity. The arbitrariness of choice might make more sense concerning a question about whether we persist with the SM + graviton approach or if we look for different proposals. I'm not involved in the world of theoretical physics so I don't know if there are genuine alternatives being actively pursued or not, but I don't see why there couldn't be. In any case, the importance of symmetry to modelling nature seems to be something about which we do not have a choice - symmetry is at work in the General Theory - so there at least I agree with you.

@StreetlightX

Not at all. The other option is simply to reject that irrationals are numbers tout court. And for the longest time this is just what happened. For a good history of this, see Daniel Heller-Roazen's The Fifth Hammer.

When it comes to how the Greeks dealt with the notion of an irrational number the term "history" is a little bit misleading I think - lack of reliable sources. There's evidence that once we get to Socrates (or at least Plato - I'm thinking of early passages in the Theatetus here) that there's no question whether they are numbers or not, just how to handle them as numbers. I've not read the work you refer to - what sources does Heller-Roazen have for indicating that the Greeks refused to regard the irrationals as numbers at all?

My claim is that all those moves which in math are universally considered to be invalid, are based in ontological principles. The principles of addition and subtraction, just like the principle of non-contradiction, is based in ontology. So consensus on these mathematical principles requires consensus on ontology. — Metaphysician Undercover

In math, the assumptions of certain ontologies (the constraints) are more likely to be agreed upon. But it is not only that, it is the fact that the subject matter is limited to specific things (numbers, relations..) which makes it more constrained from the start. Ontologies that don't take into account what numbers, relations, geometric principles do, don't get counted anyways.

one is the scope of the content of the agreement (narrow or broad), and the other is the scope of the formal aspect (the number of individual human beings engaged in the agreement). So your designation of "a higher amount of constraints' is really quite vague because it doesn't directly take into account either of these parameters. Saying that there is a high number of constraints in place doesn't say anything about the number of people engaged in each of these constraints, nor does it say anything about the scope of application of each of these constraints. For example, whether 100 instances of 20 people agreeing to some specific constraint constitutes a "higher degree of consensus" than a million people agreeing to some broad principle is highly doubtful. You cannot judge "degree of consensus" by "amount of constraints". — Metaphysician Undercover

I can't speak to the formal constraints, I can say that the content would have to be based on the quantifiable information I mentioned (relations, numeracy, geometric principles, measurement, statistics, etc.).

This extends to all forms of measurement, consensus on the unit of measure is required. This is exposed by Wittgenstein when he claims that a metre stick is both a metre, and not a metre. — Metaphysician Undercover

Yes, but I think there is more consensus in math based on the more limited scope of math.

Philosophy is less constrained than mathematics, as you say, but the constraints of mathematics are really derived from philosophical constraints. So when the philosophers produce consensus on ontological principles, mathematical constraints are derived from this. The overriding constraint, "what is valid", is a philosophical principle, not a mathematical principle. So there cannot be any consensus on particular mathematical principles without consensus on "what is valid". — Metaphysician Undercover

I agree, but because of the tendency for what is valid for what is mathematical principles, and how one can demonstrate (via proofs, via empirical evidence, etc.) then what can be demonstrated and what can be provided weight from consensus is more easily had. I cannot simply say in philosophy, "Clearly, by using one of these proofs which we all agree is a way to ensure the direction to validity in solving philosophical problems.." Nope.. maybe some camps would except this, but certainly not a universal agreement of what counts as demonstrable.

But philosophical constraints exist. And if they are not produced, or created by philosophy, where do they come from? You cannot accurately say that philosophy is unconstrained, then look at something like the law of non-contradiction, and dismiss this as non-evidential of constraint. Furthermore, what you do not seem to apprehend, is that these philosophical constraints are what bear upon the world of mathematics, as the foundation for mathematical constraints. — Metaphysician Undercover

Using your content and formal definitions of constraints- the content of philosophy simply goes beyond logical analysis (just one field of philosophy, following its own scope, assumptions, etc.). The formal definitions of constraints are also much less constrained. There are relatively fewer number of ways to show validity in math than in philosophy which can decide to straight-jacket (or not straight-jacket) its validity in any number of traditions, but they are much more varied. Validity itself can be called into question.. It is a much more fluid, constantly moving, open-ended chess game where the rules can constantly change while playing the game.

No, no, this definition of "unit" must be rejected as circular, or an infinite regress, and therefore not a definition at all. — Metaphysician Undercover

That's funny, given a circle is the most fundamentally symmetric type of unit. It stands as the limit to an infinite regress in terms of the number of sides to a regular polygon.

In any case, the importance of symmetry to modelling nature seems to be something about which we do not have a choice - symmetry is at work in the General Theory - so there at least I agree with you. — jkg20

Great.

That's funny, given a circle is the most fundamentally symmetric type of unit. It stands as the limit to an infinite regress in terms of the number of sides to a regular polygon. — apokrisis

A circle, because it has no beginning nor end, is the very same thing as an infinite regress. The circle cannot limit the number of sides to a polygon because a curved line is fundamentally different from a straight line. A polygon is made of straight lines with angels, and a circle consists of a curved line. Do you see the difference between the two, and how they are fundamentally incompatible? The curved line is not a limit to the straight line, the two are categorically different.

Consider that a circle could have whatever number of degrees we want. The choice was for 360 because it was derived from a proximity to the number of days in a year, but we could assume any number, even an infinite number of degrees to a circle. Suppose that we assume an infinite number of points around a circular line, because a line segment is infinitely divisible. Between each of these infinite number of points assume a connecting straight line. Now we have a polygon with an infinite number of sides. This is not a circle, because it is derived from an infinite number of points equidistant from a central point, rather than a circular line. So a circle does not limit the number of sides that a polygon can have. It is a completely different concept.

This is not a circle, because it is derived from an infinite number of points equidistant from a central point, rather than a circular line. — Metaphysician Undercover

The whole post was good, really good, but this is my favorite part.

Carry on.

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?

Yeah, I'll own up to being a little too oblique. I’m pretty sure we’re in agreement on the object level, I.e I get that “choice” as you’re using it means something like (very simplified): the existing field of concepts + what (new thing)you’re trying to do with them determine new conceptual moves. This as opposed to progressively capturing broader swathes of some pre-existent truth. And you can’t just make up whatever rules, because you’re always already operating from within a dense conceptual web. And then there’s the T S Eliot-esque thing about how new choices retroactively reconfigure the pre-existing field. (The sense is always open)

The meta thing is more a kind of frustration about how this kind of analysis has itself become a kind of dogma (in certain quarters.) Dogma’s probably too strong. It’s become the central focus. It’s a step past Derrida (as stereotyped) because it’s concerned with creative construction, rather than deconstructive handwringing in the face of the void left by metaphysics. But, there’s no way around this, once it becomes a scholarly debate around an invariant process by which interventions in conceptual space are made, than this begins to calcify into its own kind of sub species aeternitatis. You don’t have some global theological container, but you do have a invariantly structured engine (with a leap, or a shove, at its heart). I think this is as theological as anything else. It’s tribute paid in ornate paeans to the generative moment. So pagan, rather than Christian, but theological nonetheless.

I’m not slamming the analysis at all, only it seems like analysis you’ve already done. It seems like a retracing. That’s what I was hinting at with the hamlet thing.

Thanks Srap. This is really the difference between continuity and discrete points. You could construct a circle with points equidistant from a centre point, but this circle would be lacking in continuity, the points being discrete. If you connect the points, you cannot connect them with a straight line or else you do not have a circle, you have straight lines at an angle to each other. It is this queer aspect of the circle, that the continuity from one point to the next is not a straight line, which makes pi an irrational ratio. The continuity between one sot on the circle and the next, must always be two dimensional and cannot be represented as a one dimensioned straight line. The diameter of a circle is a straight line (one dimensional), while the circumference is curved (two dimensional), so the two cannot be measured with the same units of measurement.

You cannot measure a two dimensional object with the same units of measurement as you measure a one dimensional object because you need to allow for the defined difference between the dimensions. This is expressed as the angle. The relation between a first dimension and a second dimension is described by Euclid with the parallel postulate. The concept of "angle" is required to relate two distinct dimensions So if you measure a one dimensional object (straight line) with a specific unit of measurement, you cannot measure a two dimensional object (arc, or circle) with the same unit of measurement because the angle, which allows for two dimensions rather than one, needs to be accounted for and cannot be measured with that unit of measurement. Angles are measured by another unit.

The curved line is not a limit to the straight line, the two are categorically different. — Metaphysician Undercover

Hmm. What then of the points that make the circle. Are they not the smallest possible straight edges?

A point is the limit to a line - the zero-D terminus that has greater local symmetry than the 1D line which is having its own symmetry broken by being cut ever shorter, and eventually, infinitely short. A point is simply a line that can't be cut any shorter.

Then for a line to be either straight or curved is itself a question embedded in the 2D of a plane at a minimum. So curvature, or its lack, is determined by the symmetry breaking of a more global (2D) context. A line becomes "straight" as now the locally symmetric terminus of all possible linear wigglings.

Straightness is defined in terms of the least action principle. A straight line is the shortest distance to connect two points. You may be familiar with that story from physics.

So now what can we rightfully say about the points that make up the circumference of your circle?

They are minimal length lines. But are they straight or are they curved? Or would you say the issue is logically vague - the PNC does not apply? No wiggling means no case to answer on that score.

Anyway, it is clear that the straightest line is simply the shortest path in regard to some embedding context. And even you would agree that a circle is composed of points. So the standard view - that a circle is the limit case, an infinite sided regular polygon - holds.

Thanks Srap. — Metaphysician Undercover

You're welcome?!

t’s a step past Derrida (as stereotyped) because it’s concerned with creative construction, rather than deconstructive handwringing in the face of the void left by metaphysics. — csalisbury

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.

Is this academic? I mean, yeah a little, especially since I keep circling back to 'philosophy' as my object of analysis, rather than anything else in particular. And I admit that that's out of a sense of comfort and ease, soothed also by the fact that the individuation of philosophical concepts is no different to the individuation of anything else in the world. So the 'next step' is to try and think about all these ideas in terms of 'worldly things', bodies, ecologies, economies, political organization, etc, etc. But that shit's really hard to do, and I'm still (obviously) wrangling with the basics. I mean, at least give me credit for bringing the math into it, I thought that was pretty novel, even for me. But yeah, okay, paganism, you're not entirely wrong, but then, I'm still working through shit man, and I'm allowed to do it because you're not my thesis advisor and I'm taking the leisurely route and attack a small area from different angles even if you think it's all a bit samey.

I mean, yeah a little, especially since I keep circling back to 'philosophy' as my object of analysis, rather than anything else in particular. And I admit that that's out of a sense of comfort and ease, soothed also by the fact that the individuation of philosophical concepts is no different to the individuation of anything else in the world. — StreetlightX

Except what is demonstrable in the philosophy world cannot be considered valid simply by proof as in the math world, which has the luxury of consensus as to types of proofs. Philosophy only has this within their camps, if at all. It is a moving target. Thus, in math the answers are more dictated by its antecedent theorems, and a more limited number of ways to frame a question, unlike philosophy. Perhaps we can debate the nature of proofs and their relationship to logic, and logic's relationship to philosophical arguments.

I don't know how else to explain that I'm concerned with concepts and not proofs. This is the third time now, and you keep talking about something else.

I don't know how else to explain that I'm concerned with concepts and not proofs. This is the third time now, and you keep talking about something else. — StreetlightX

I guess I am caught up in moving from one concept within the field over another. How one moves from rationals only to irrational, from Newtonian to non-Euclidean, choosing between category or set theory, etc. The fact in math how some concepts can overtake (as more accurate) than previous versions, and this can be agreed upon by the math community. This doesn't happen in philosophy. Perhaps, this leads to a more abstract notion as well that in math, there is a sort of determinism because of the constraints that dictates the possible next moves. So I guess the concept overtaking/consensus is more to do with the pragmatics of the math concept formation, and the concept of the constraints dictating the possibilities is more epistemological.

The fact in math how some concepts can overtake (as more accurate) than previous versions, and this can be agreed upon by the math community. — schopenhauer1

No, wrong. Explained already.

No, wrong. Explained already. — StreetlightX

Not really. You discussed things like the difference between category and set theory, but in mathematics, it is possible that there will be a consensus that one is more accurate than the other. In philosophy not so much.

No, 'accuracy' is not at issue, never was, never will be, not even in principle.

You said:

And they did. They gave up (2) - the idea that all numbers were expressible as ratios. In doing so, they expanded and changed the definition of number. Now, numbers included both rational and irrational numbers, where they didn't before. Moreover, they no longer were measures of length (Note that this was not an easy choice for the Greeks to make. Legend has it that Pythagoras - or his followers - sentenced the student who discovered the irrationals to death by drowning: such was the heresy of a non-rational number).

So what's the moral of this story? Well, for B&C, the important point to note is that nothing in the math itself forced this choice, rather than the other. Rather, the choice was made on the basis of 'extra-mathematical' considerations: giving up (2) would allow us to take measurements of things like the diagonal of right-angled triangle ( = √2 = 1.4142... etc). Here is how they put it: "The choice between criteria, whatever its motivation, does not answer uniquely to intra-mathematical considerations; mathematics itself, we might say, allows either choice, while eventually accepting the choice that is made." And as they go on to detail, the history of math is full of these decision points, imposed by the math, but not decidable by it. — StreetlightX

The choice as not arbitrary.. giving up 2 essentially forced their hand on this if they were to move forward with answering questions of non-fractional numbers. Eventually a consensus forms as to what counts as more accurate.

Edit: And indeed it has everything to do with this.. it is just something you are overlooking- the pragmatics of consensus in math vs. philosophy and the dictates of mathematical reasoning in forcing a decision to the one that seems to fit the models/demonstration.

No, as to what counts as more useful. That one needs a lever to move a weight does not make the lever 'true'. That's just bad grammar.

No, as to what counts as more useful. That one needs a lever to move a weight does not make the lever 'true'. That's just bad grammar. — StreetlightX

But the model that models the lever's action through proof each time is not. Two models are used.. eventually in math, one might have a consensus as the more accurate model.

Maybe, maybe not, but that's just displacing the conversation again; you're talking about something else. That's what I keep tying to tell you.

you're talking about something else. That's what I keep tying to tell you. — StreetlightX

Well, maybe I am then if you say so..It's your thread, and I am just trying to add what I thought was something overlooked. Where you see math as not being dictated by the internal mathematics itself, I do see this, through the constraints of what math is trying to investigate. Where you see a sort of contingency in picking certain ways of looking at math, I see a sort of determinism albeit one that is manifested through consensus in the mathematics community.

'Next-step analysis', the kind I'm not ready to do, even though still general: "Nevertheless, from the moment when capitalism crystallised, when it occurred, it unleashed a necessary and constraining process. This is why history must be theorised as a mixture of the aleatory and the necessary, proceeding by contingent leaps, determining a problem that retroactively produces a prior history without emerging from it. ... What is at stake is a transductive crystallisation in the strict sense that Simondon uses it, even though he did not think of applying this concept to the history of capitalism: a contingent irruption (chance) unleashes its own logic, its virtual problem, from which the supposed linearity of prior history is retrospectively configured. This is why ‘there is no good reason but contingent reason; there is no universal history except of contingency...

...This is why, in the third chapter of Anti-Oedipus, Deleuze and Guattari write that there is no universal history except that of capitalism. This does not mean that capitalism is inscribed in an all-encompassing, universal history, but, on the contrary, that it creates the conditions of possibility of such a history: there is no universal history without capitalism. There is no universal history of civilisations except that of capitalism, just as there is no universal history of Reason except that of philosophy. It is not that the two determinations are teleologically programmed in advance, but rather that from the moment when they occurred (as a contingency), their emergence retrospectively unifies all prior attempts, through the construction of the universal. From the moment when capitalism emerged, it unified prior histories because it configured itself as universal, acting pragmatically as an instance of domination. But this universal is contingent (it is not necessary that it crystallises at a particular moment)." (Anne Sauvagnargues, Artmachines)

The history of math proceeds in this way too...

It is not that the two determinations are teleologically programmed in advance, but rather that from the moment when they occurred (as a contingency), their emergence retrospectively unifies all prior attempts, through the construction of the universal. — Artmachines

PoMo is going to get such a shock when it catches up to 1980s work on universality in dynamical systems. :)

In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system.

https://en.wikipedia.org/wiki/Universality_(dynamical_systems)

A word on proofs: proofs (and with them, truth) are trivial, they're bureaucratic grunt work, the labour of pencil-pushing; proofs are only ever the product of getting the concepts right in the first place, of drawing up the right kinds of distinctions and correctly distributing the right categorizations of sense. Consider the recent proof of the Gaussian Correlation Inequality, as detailed in Quanta. It discusses a problem more than half a century old, recently solved.

The relevant question is: what did the 'solving' involve? A: It involved forumlating the problem correctly, posing it correctly: "Royen hadn’t given the Gaussian correlation inequality much thought before the “raw idea” for how to prove it came to him over the bathroom sink. ... In July 2014, still at work on his formulas as a 67-year-old retiree, Royen found that the GCI could be extended into a statement about statistical distributions he had long specialized in. On the morning of the 17th, he saw how to calculate a key derivative for this extended GCI that unlocked the proof. “The evening of this day, my first draft of the proof was written,” he said.

...Over the decades other experts had been attacking the GCI with increasingly sophisticated mathematical methods, certain that bold new ideas in convex geometry, probability theory or analysis would be needed to prove it. Some mathematicians, after years of toiling in vain, had come to suspect the inequality was actually false. In the end, though, Royen’s proof was short and simple, filling just a few pages and using only classic techniques. Richards was shocked that he and everyone else had missed it." Note here that the creative element, the innovatory aspect occurs before a single line of the proof is formulated, a proof which took him a day or so to write.

People have this incredibly naive idea that 'problems' (not just in philosophy, but also and especially in math) are always well-defined and it's all just a matter of going through the motions to find the right answer. But the 'answer' is almost always a case of getting the problem itself right, of understanding the very nature of the problem at hand; proofs 'fall out' of this prior conceptual work, which is where the real effort of grappling with the problem lies; proofs are the crust of bubbles on a wave, they are everything that is unimportant and trivial about solving a problem because the 'solution' to a true problem is always in the very way the problem is articulated. As a philosopher I was reading recently put it, "[Truth is] the outcome of an infinite process of sense-production. Truth is the limit object of a production of sense."

--

Fdrake was also tellinng me about the proof for the 'Classification of Finite Simple Groups', which apparently is 17 volumes long, and which is one of the most important theorems in all of statistics. I'll simply quote what he wrote to me about it: "During its finalisation tonnes of slips in logic and rigour were found, only one or two blokes who served as the coordinators for the research program of finishing the book knew how to direct things globally, to this day (afaik) no one has a comprehensive understanding of the proof in every detail. It would take a lifetime, then you'd die. The best part of the story is that no (exaggerating here) pure mathematicians doubted the whole theorem because they already knew the idea was the right one and the logic was almost an irrelevant detail. These kinds of details gets derisively called 'casework' a lot of the time."

Mm, a shock indeed that anyone could be so philosophically backward in their interpretation of the results.