Math and Motive
In their wonderful paper on Wittgenstein and math (found here), Michael Beaney and Robert Clark (B&C) note that the history of mathematics has continually confronted us with choices - choices which, once made, cannot be reconciled. They give a few examples, but I'll focus on the simplest one for the sake of space: the choice the ancient Greeks had to make concerning the definition of a 'number'. Prior to the discovery of irrational numbers (numbers that cannot be expressed by ratios, like ?), the Greeks understood numbers to be both:
(1) Measures of length (every number corresponds to a measurable length, like a table-leg) and
(2) Expressible as ratios ('every number can be expressed by a ratio, like x/y').
The discovery of irrational numbers - numbers that can't be expressed by a ratio - meant that at least one of these criteria had to be given up. So the Greeks had to make a choice. 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.
(Two further quick examples for the mathematically literate: the 'choice' to allow imaginary numbers (the square root of negative numbers, like ?-1), or the 'choice' to give-up the postulate of parallelism in geometry, allowing for the development of non-Euclidean geometry).
For B&C, the important point is that the choices made, although forced by the math itself, are nonetheless grounded in what we aim to do with the math, considerations which are not dictated by the math itself ('extra-mathematical'); if we want to measure the hypotenuse of a right-triangle, then we must - by necessity - drop the second criterion. Otherwise, we are free to keep it. What I want to add to this is that philosophical concepts are just like this. The concepts we employ are a function of what we aim to capture with them; to employ one concept rather than another is to bring out one aspect of the world rather than another. Moreover, the deployment of our concepts is not governed by truth, but by their range of illumination. This is not on account of their being arbitrary ('subjective'), but absolutely necessary.
(1) Measures of length (every number corresponds to a measurable length, like a table-leg) and
(2) Expressible as ratios ('every number can be expressed by a ratio, like x/y').
The discovery of irrational numbers - numbers that can't be expressed by a ratio - meant that at least one of these criteria had to be given up. So the Greeks had to make a choice. 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.
(Two further quick examples for the mathematically literate: the 'choice' to allow imaginary numbers (the square root of negative numbers, like ?-1), or the 'choice' to give-up the postulate of parallelism in geometry, allowing for the development of non-Euclidean geometry).
For B&C, the important point is that the choices made, although forced by the math itself, are nonetheless grounded in what we aim to do with the math, considerations which are not dictated by the math itself ('extra-mathematical'); if we want to measure the hypotenuse of a right-triangle, then we must - by necessity - drop the second criterion. Otherwise, we are free to keep it. What I want to add to this is that philosophical concepts are just like this. The concepts we employ are a function of what we aim to capture with them; to employ one concept rather than another is to bring out one aspect of the world rather than another. Moreover, the deployment of our concepts is not governed by truth, but by their range of illumination. This is not on account of their being arbitrary ('subjective'), but absolutely necessary.
Comments (243)
Math-games, eh? But is there one math-game to rule them all?
I would say there is a difference here between math and philosophy then based on what you present. Math dictates (by discovery) that a different method be used to solve a problem. In philosophy, there are no clear-cut discoverable rules that can only be used to answer a particular problem. Rather, any number of worldviews/heuristics/logical constructs can be employed to answer the philosophical problem at hand. The math "dictates" that a particular rule must be used, philosophical problems have no demand. If the mathematician has a hang-up on a particular math that "works" for that problem, that is their decision, but they will probably fail to answer the question at hand based on a bias.
The literal point of the thread is that the math dictates nothing about the choices that must be made. The problems dictate the directions into which we take math; and we are motivated by problems (there are of course intra-mathematical problems, but these too are no different (cf. Rosen on Modelling).
Yes, then change what I said to the "problems of math", I don't see how this counters the argument. If a problem requires a new method in math to solve a problem, it cannot be disputed that the math solves the problem if it "works". Pythagoras for example, didn't like irrationals, but it worked to solve problems that when worked out can clearly be seen by all. Philosophy has no clear-cut agreement whether the problem "works". If another method follows the same problem, then both can be said to "work", then it is preference for which is easier for that person to use. What cannot be contested is that a method(s) is agreed upon to work in a mathematical community. Philosophy can never really have this definitive satisfaction that their solutions "work". So one illumination is agreed upon, where in the other camp the illumination is a continuous dialogue. The discovery aspect to math is the difference. You cannot stretch this to philosophy no matter how you want it to be so.
But I'm not talking about the problems of math. At least, not exclusively. So there's no good reason to make any such change.
Further, among the points that B&C stress is that it is not at all 'discovery' that is at stake, but what they call - following Wittgenstein - concept-determination: "what is going on here is best described neither as ‘discovery’ nor as ‘invention’ of something entirely new. There are facts to be revealed, and creativity to be exhibited, but what is crucial is the opening up of different aspects of something ... which prompts a choice that sooner or later ‘catches on’... and proves fruitful."
This indeterminacy actually shows up as a methodological problem in stats, called researcher degrees of freedom.
But you compared the problems of math to the problems of philosophy, so I am just taking your cue.
Quoting StreetlightX
And the point I am trying to make contra your comparison is that while both might have an "opening up of different aspects of something..which catches on", the "proves fruitful" part is what is different between the two. In philosophy, there can never be a set agreement that "this works" to solve a particular philosophical problem (at least perhaps outside of logic...something akin to math). However, in the math community "this works" can be agreed upon. So "proves fruitful" here is not just a difference of degree but quality. Fruitful in math really cannot be disputed- everytime one does the problem it works out. If this method is overtaken by a better one, then that one will be agreed upon to work out, in a step-wise fashion. If both methods work for the same problem, then it is preference perhaps. However, in philosophical problems, it is always open-ended. Fruitful here can be continuous, ongoing, etc. It is more invented, and not dictated by the rules of the game as much as math which is very much driven by the results that are calculated- what are the constricted givens of the math world.
Edit: So I guess, there is more "imposed" then is indicated from the OP about math. One can find a new math ugly, not to one's liking, but it cannot be argued that the results work. Eventually, this forces one's hand. Not so much in philosophy.
My point earlier to StreetlightX was that a difference between math and philosophy on new insights or "illuminations" is that math must eventually acquiesce to what works to get the results. Philosophy can keep going on open-ended infinitum. 0 was eventually incorporated because it worked and everyone could see the results when it was used. The results demands or "dictates" its eventual use- even if it is at first ignored or scorned. No one for example will say, "Hey we now all acknowledge the Age of Schopenhauerean Ethics" when solving ethical problems. As much as I'd like to see that, it ain't gonna happen.
Irrational numbers - following the Aristotelian argument supplied in the paper - can be seen as having to relax the constraint that a number is either odd or even. The argument shows that an irrational number is not definitely one or the other. So a higher degree of symmetry is obtained by removing this constraint as an "obvious necessity".
The same goes for complex numbers - removing the constraint that number lines be one dimensional. And non-Euclidean geometry - removing the constraint that worlds be flat thus parallel lines apply.
Quoting StreetlightX
No. The tension here was between the numerical and geometric view - the geometric one representing the presumed continuity of physical nature, the numerical one representing the desire to talk about that in a system of discrete signs.
So you have the continuity of symmetry and the discreteness or individuation that results from symmetry-breaking. And then the metaphysical tension regarding whether to understand acts of individuation as actual discrete ruptures - stand alone existences - or merely contextualised developments, breakings that have only relatively definite existence.
The disturbing thing was that geometry and algebra did seem to be two commensurate or complementary ways of talking about the same world. But eventually the cracks got revealed.
Numbers work as zero-dimensional and non-geometric signs when their physical dimensionality is suppressed or constrained. But to maintain a complementarity between numbers and geometry as maths seeks to advance, gradually the constraints on that dimensionality have to be relaxed in a systematic fashion to keep the two worlds connected as we move deeper towards maths capable of higher symmetry.
So in summary, it does all start in pragmatic acts of measurement - the semiotic trick of giving names to things. We break the continuity of physical experience by imposing a system of discrete marks upon it.
This is the useful trick. The semiotic ability to construct constraints that break the symmetry of the experienced world. And the Ancient Greeks were dazzled by this new Platonic reality that rational geometry opened up. A fundamental connection between discrete number and continuous dimension appeared to be forged.
And gradually the very nature of that trick - even for the physical world itself, the pan-semiotic Cosmos - got revealed. The whole damn world exists as a constraint on dimensionality, a grand tale of global symmetry-breaking and localised individuation.
So maths only ever had one route by which to recover its physical origins. At every turn, it needed to work out what further localising constraint had been added to the deal that could be successfully generalised away. Mathematical advance had to see that a symmetry had been broken, which could then be unbroken to reveal the higher, more abstracted, less particularised, realm that lay beyond.
Go up a level and you could still give names to things. Topology abandoned geometry's definite measurements of length, but still deals in individuated abstracta. Semiotics could still work as the trick.
But the route was always Platonically predestined and necessary. If existence takes definite shape due to constraints, due to symmetry-breakings, then the only way to understand that is by following the path backwards that abstracts away those constraints, unbreaks those symmetries, to reveal how the how show works.
Justly so. I think we ought to recall the sense of awe with which the ancients treated mathematics and reason. After all, at that point in civilisation, life truly was 'nasty, brutish and short' for the great mass of people. But then early civilisations also witnessed the construction of the pyramids, the discovery of astronomical tables, and many other such principles. (The story of Archimedes' achievements is quite amazing. Not to mention the Antikythera device.)
So, wasn't it only natural that the power of reason, and the uncanny efficacy of geometry and arithmetic for the construction of such apparent miracles, would be regarded with a kind of awe? It was what separated humans from beasts, and also philosophers from the hoi polloi; the 'rational animal' alone could fathom such principles. Hence the later identification of 'the rational intellect' and the soul.
Quoting Wayfarer
We have numerous choices as to what zero actually represents, and this is evident if we start to look at the difference between the different representations of zero, like the number line, in which zero represents a count of one unit, or in some cases zero represents the point of separation between positive and negative. Then there is the representation of imaginary numbers, in which I don't quite know what zero represents.
I disagree. While I think the whole symmetry-breaking story is a useful framing and pedagogic tool - I turn to it too, occasionally - I think it is a mistake to reify it into a metaphysical picture. It ends up treating the pragmatics as mere accidents on the way to some eternal Platonic story which was there from the beginning - which is nothing more than theology through and through. But, to borrow a bit from Rosen - I don't believe there is any 'largest model' here: there is no 'most general generality' - what I suppose you call 'vagueness'.
Vagueness is for me the ultimate transcendental illusion: it takes a perfectly valid move - the step from particular to general, always motivated by a particular problem (B&C's 'decision points') - and then illegitimately extrapolates that step into what one might call an 'unmotivated generality'. It tries to think an abstract generality shorn of any reference to the particular, cutting off it's roots to any particular problem that would motivate it - other than a nice, just-so story. Abstraction without a (necessary) foot in the real.
So basically I can agree with you up right up until the point where you invoke unmotivated generality as a Platonic bow to tie the whole developmental story together. It's this very last step that shifts a perfectly rigorous and valid methodology into a procrustean metaphysics that tries to retroactively fit concrete developments into a pre-ordained story. It's just a theological-Platonic hangover/residue that needs to be rejected.
I think this is almost right in a few ways, but when you say "their range of illumination" I'm reminded of the old adage about looking by the lamppost for the lost keys at night.
IOW, in venturing beyond the known, we do in fact use truth as a guide - in the sense of it being an ideal to aim towards.
I think you're missing the point though - what 'proves fruitful' is the choice made between two possible 'paths'. We're not talking about 'solving problems': we're talking about determining concepts: should number be treated in this way or that? Should infinity be thought of like this or like that? The point is that the normative force of this 'should' is provided by a concrete problem (which may be intra-mathematical or not) which any choice that is made is responsive to. Mutatis mutandis the way in which we form concepts in philosophy are similarly responsive to the problems they address: in neither case is it a matter of solving problems, but determining concepts.
The two criteria the authors mention need actually be unpacked as the following (as far as I can see):
1) There is a one-one correspondence between numbers and lengths and there is an arbitraily chosen unit length.
2) Numbers can be expressed as ratios of the varying amounts of the arbitrarily chosen unit length mention in (1).
Is there really a choice as to which one to give up? Giving up 1) would be to give up using the notion of a unit of length, which would entail nothing could be measured.
I believe the idea is that it is no longer part of the definition of number that it be a measure - which doesn't mean it can't be used for measuring.
So from the B&C article itself it says:
[quote=B&C]We now take all this for granted, but if we go back to the origin of the
determination, we can see that it was by no means necessary. At the core of this
determination was a choice of conceptual aspect, and although we might find it hard
now to see things in any other way, it is important to recognize that the choice was
there and that our concept of number might have developed in another way. We
should also note that while the choice between the different ways of seeing – of
determining the concept – was, we might say, forced by mathematics itself (the proof
above), the outcome of the choice was not so determined. 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.[/quote]
Again, I don't discount that there is some sort of choice as to which criteria fits the picture better, but rather, I disagree with the move to equate it with philosophical creativity. In the math world (similar to the science world), eventually there will be a consensus (for a time being) of the criteria. This consensus is based on the fact that it can be used to solve a wide range of problems demonstrably. This demonstrable ability of math is lacking in philosophy. There is a sort of constraint or "dictate" going on here in math, moving it along to a more clear picture, that is not able to be had in philosophical problems. Philosophical problems are more like interesting flourishes of thought. Whether math has necessity or not, the problems are constrained enough to have its own dictates through demonstration. Philosophy does not. They are flights of fancy, if you will, that can be entertained or not entertained with no demonstrable constraints on the flights of fancy one chooses. It is too open-ended for any consensus. There is no "pairing off" of previous notions in a step-wise fashion.
I suppose then that we simply disagree on this point. I'm firmly of the belief that every philosophy worth its salt has the kind of internal consistency that characterizes mathematical concepts, and they derive that consistency from the particular problems that animate them. The difference, to the degree there is one, lies only in the fact that philosophy has a far wider range of inspiration than math: its problems are drawn from a more diverse array of sources. To dismiss philosophy as 'flights of fancy' is to not understand it.
To this I'll admit, I should have used another word. Perhaps "thought-explorations" or "generalized theoretical-investigations".
Quoting StreetlightX
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, there can be no consensus really except perhaps those who are specialized in that particular philosophical school of thought. So maybe Wittgensteinians agree on certain principles and can branch out from there, or Schopenhauerians can in their camp. But then, this just goes to my point that in math there is no "school of thought" in math-proper. It is consensus of concepts that are demonstrable in the results they produce and works in a step-wise fashion. I guess my two main terms to see the difference between math and philosophy are "demonstrable" (which leads to eventually) "consensus" (and onward it goes).
Quoting StreetlightX
But this is a much larger difference than you seem to be implying. The implication with math is that its constrained world "dictates" its next move, where in philosophy the diverse array of resources makes it too open-ended.
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.
p-hacking isn't really like that. The damndest thing about it is that it's almost always well motivated. If you have a dataset trying to study some frontier phenomenon you're going to explore it with as many models and quantification procedures that make sense as you can; you're also going to throw a lot of scientific hypotheses and statistical sub-hypotheses at it.
The danger in p-hacking isn't an inherent feature of the p-value (which was originally proposed as an exploratory tool with no thresholding values like 0.05), it arises more from the incentives for researchers to find sexy publishable conclusions from small datasets and too much confidence in pilot studies.
So, it's not really that it's cynical (most of the time), it's just the way people treat statistical analyses; like they're literally taught p<0.05 = publishable result; and are made to by the publish or perish sexy things doctrine. Rather than pre-registered replication studies and data analysis/dataset sharing after anonymising - this robustification of science isn't incentivised at all.
So to link it back to the OP: a lot of the problems in philosophy probably also come back to asking the wrong questions. But it isn't like the wrong questions are often asked with an agenda, it's stumbling around the garden of forking paths without a map; that map's probably some knowledge of philosophical history surrounding the problem and reading it's a series creative leaps of rationality.
Not really. If we are talking about a pan-semiotic metaphysics now, the goal is to divide reality into its necessities and its accidents. So pragmatism is about some finality being in play and shaping events in a contextual fashion. Things are individuated by constraints as a matter of top-down necessity. But constraints themselves are open or permissive. They only limit to the degree it matters. Beyond that, the accidental or spontaneous can be the rule. Constraints are only concerned by the differences that would make a difference.
So in terms of metaphysics, the question becomes what is the most universal goal? And one obviously sensible answer is the limitation of instability. If any kind of world is going to exist - given the primal nature of chaotic action - then it has to develop the kind of regularity that gives self-perpetuating stability.
Thus the "Platonic universals" would be an evolutionary story. The need for stability would be selected for just because stability is definitional of what we mean by existing. It would be the first necessity. And the eternality of that form would be an emergent and immanent fact. It would be what gets expressed by the end in the long run.
And then, some kind of stable world having established, this could be the ground for the development of further, more particular, states of constraint. Other more localised purposes, expressed by more complex forms, could arise - stabilising "sub-worlds".
So it would be pragmatics all the way. Constraints to limit dynamics and produce particular states of entification could keep developing in open-ended fashion. In the most universal view, they would be accidental - differences that don't make a difference to the most universal view. But locally, in being further hierarchical levels of development, they would encode the forms that are necessary to that further level of organised and individuated being.
That is the metaphysical picture. Maths then becomes the science that explores the principles of form. It seeks out the structures, the rules, that stand for the necessities - with the accidental part of reality becoming whatever measured values we might insert into some general rule.
Quoting StreetlightX
I wasn't talking about vagueness in reference to mathematical notions of symmetry. I was talking about the continuity of generality quite specifically.
In Peircean logic and metaphysics, the particular and the general co-arise from the vague. So vagueness is firstness, then particularity would be secondness, and generality is thirdness.
Sure, that makes vagueness a kind of ultimate symmetry - it gets broken by that definite division into the general vs the particular. But here, for the moment, the focus is on the ontic structuralism story - the general or universal constraint that "being intelligible" has on shaping the particular or the local. That is the territory that maths is exploring by abstracting to discover the most generalised forms that limitations can intelligibly take. The question being asked is what is the most primal kind of individuation.
And given your interest in individuation and contextuality, it is odd that you don't see this immediately. The step from the particular to the general is about discovering what context of constraint is causing the particular to be what it is in the first place. From there, one can start to remove those constraints - in a most general fashion - to get down to what is most primal about individuation itself.
It is in generalising away the constraints that individuate that we arrive back at vagueness as a formless and unindividuated ground. But that vagueness is a material potential. And maths focuses on the issue of the forms that could organise that. That is why maths does not talk about energy, just organisation. That is why maths ends up talking in "Platonic fashion" about the finalities that want to be expressed.
For there to be (persistent) individuation, there needs to be (embodied) constraints. So reality has to be actually organised as a material system. But maths targets just the constraints, understood in terms of logical generalities. It imagines them having their own abstracted existence - as a ghostly organising hand. And it is healthy to do this as it is the way that it can focus on what is essential vs what is accidental in our accounts of nature. Pragmatism relies on being able to know the difference - the differences that make a difference vs the differences that don't.
Quoting StreetlightX
But this is just you forcing things into a framework you feel you can quickly reject. It's not the story I tell.
Remember the evolutionary principle at the heart of this. For anything to exist in persistent and individuated fashion, it must mean that some primal state of constraint managed to work. Everything else then follows naturally from the fact that instability could develop limitation.
I think that the premise of B&C is a little inaccurate with (1). The basis of the number system, and the foundation for Pythagorean idealism and Platonism is that the numbers signify units, not necessarily units of length. I believe Plato describes how the Pythagoreans held that unity was the fundamental concept of mathematics, and the concept of numbers is developed from the distinction between one and many. The ordering between one and many is not a matter of choice, because we cannot count backward from an infinite number, to one.
Number is a measure, but not necessarily a measure of length, and the B&C article assumes number to be a measure of length, not a measure of the difference between one and many.
This is an important point to uphold, because the incommensurability which gives rise to the irrational numbers is only produced when the numbered units are units of measurement. This implies that the incommensurability is not a feature of the numbering system itself, but of how the numbers are applied toward measuring dimensional space. The incommensurability between the two perpendicular sides of a square (the square root of two), probably really indicates an incommensurability between the two distinct dimensions of space.
However, the B&C problem can be reintroduced in a much more comprehensive way by considering the introduction of zero into the numbering system. The zero acts to negate the unit, and all units, such that if we assume that the numbering system is based zero then it is no longer based in the distinction between one and many. Zero is more like the potential for unity, or units. From the assumption of zero we have all sorts of choices for defining unity and ordering the one and the many. The proper inquiry might be as to whether numbers should be based in zero, or in the distinction between one and many. The former giving us choice, the latter necessity. The problem with starting at zero is that we have no rational way to produce units from nothing, so zero cannot represent nothing, in any real way, for the production of a numbering system. Therefore we must determine what zero actually represents.
Quoting StreetlightX
Aristotle resolves the issue of discovery/invention with the distinction between actual and potential. The act of the mind of the geometer who "discovers" the principles through constructing geometrical figures, actualizes these mathematical principles. If they exist prior to being actualized by a mind, they exist in potential only. He then uses the cosmological argument which demonstrates that nothing potential could be eternal, in his famous refutation of Pythagorean and Platonist idealism which insists that these principles existing separate from human minds, must be eternal.
It follows: The philosophy of choices is itself an example of the thing the philosophy of choices describes.
[quote=street]The concepts we employ are a function of what we aim to capture with them; to employ one concept rather than another is to bring out one aspect of the world rather than another. Moreover, the deployment of our concepts is not governed by truth, but by their range of illumination[/quote]
What is being illuminated here? To take a well-worn example: We understand & thematize the hammer only when it breaks. The implicit becomes explicit when we can no longer unthinkingly rely upon it, and so have to explain it. What's broken? What old implicit is being forced into explication?
[quote=sx]For B&C, the important point is that the choices made, although forced by the math itself, are nonetheless grounded in what we aim to do with the math, considerations which are not dictated by the math itself ('extra-mathematical')[/quote]
The motivating problem, for the philosophy of choices, is the problem that we don't know what we want to do. We can't choose, we can't decide. So the moment of choice becomes the object. We show that we, non-choosers, know about choosing better than any of the people who ever chose.
@Srap Tasmaner is right to focus on the threat of relativism. The focus on frames, versus what is framed, threatens to fling us into anything-goes. How do we choose the frame?
The solution offered, the quasi-badiouan one that truth still exists, but in the problem itself - that isn't really an answer. Or another way to say it: as it answer it's function is this:
[quote=street]will respond tomorrow[/quote]
This sums it up perfectly. The philosophy of choice is a hamlet philosophy. (Deleuze knew this.) It wants to do the famous monologue without avenging the crime that elicited the monologue. And it's done it for so long, now, that it doesn't even know what the crime is. "The truth is in the problem" is an IOU full stop.
A problem of knowledge? No. A problem of life, a problem of living. You're missing the empiricism. I believe in tomorrow. Do you?
To take a graphic example: Bimbofication, a fetish subculture involving plastic surgery and intentional self-dumbing down in order to meet a 'bimbo' ideal. It's totally a line of flight. Accept this self-modification and that self-modification and then go wild. Only it really hurts people, and also *marks* people in a way that make its difficult for them to then get out of it, and still be taken seriously as a person. (nb I'm not talking about whatever tattoos or piercings. I'm talking about body-modifications done specifically for this subculture)
So then I want to say: No, not *that* kind of line of flight. But, yes, picking a frame that fosters a (somehow spiritually or intellectually nourishing) line of flight.
But then that already is outside of frames and into *values*, which I think is what's missing here.
The major point being: I think choices/decisions involve the whole heft of your spiritual being. I don't think lines of flight always do. They can funnel you into addiction, or some kind of subculture crutch. But I think the missing element here is 'values' which is what the 'truth of problems' hints at.
Eh. Looks like frames all the way down to me. Wouldn't be so many philosophies of the event otherwise. I don't think you can escape the regress that's ultimately truncated through what you do; that's what it means to make your mark.
I think that's what I mean though. 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'. (For example: Mao as Badiou's truth-event, but Badiou wasn't an actor in the cultural revolution. You could, cheekily, call this a kind of 'orientalism' and I think you'd be right.)
I agree!
I don't agree with the first bit. In my book perspectives accompany frames; occupying a frame is characterised by how you navigate in and out of it; moving in and out of it in any way changes the other frames; I think of it like relative motion. Though, the usual way people occupy frames constrains variation in their own frame changes by a delimitation of how the other frames are embedded perspectivally into each other. Most don't matter, some matter a lot, sometimes we're surprised by something that didn't matter becoming something (or already was something) that matters a lot.
Big D decisions are aligned with stuff already mattering a lot or stuff coming to matter a lot. First's a perturbation in stance on stuff in general; like a personality or value system, it's an island of sense demarcating what's nonsense. So it looks intrinsic, and is intrinsic to a frame for most intents and purposes. The first one is also usually accompanied by some combination of volition, permission and dedication; I choose to quit smoking as a frame (big D) every time I refuse a fag (little d). Another way of putting it is it's the conditions that naturally accompany the frame. Big D decisions in the context of little e events.
Then you get that second type, big D decisions associated with big E events. Those don't happen very often. In early Heidegger's dreams authenticity is aligned with impossibly choosing your big D and big E. In Badiou's it's events leaving subjectivities behind; the impossible for X forcing itself upon X in the big D decision of accommodating or resisting it.
Edit; so I think of people and values as somewhat multithreaded or parallelised, people generally live in the context of big D decisions living little e events consistent with them. Sometimes catastrophe or joyous novelty happens and you get a fuckoff big E to fuck with your D. What counts as a big E makes sense in the context of a big D.
Edit2: in the context of the thread, this has a lot to do with Badiou's 'fidelity to the event', when events are problems. You let that problem restructure stuff to reveal the problem's essential nature, then address it. What's the essential nature? I guess some highly active frame making waves of nonsense drown people sleeping on the beach in their islands of sense.
Edit3: underlying some of this is me working through thinking about the body, affects and sensations as aligned with e's rather than d's.
*Edit: Problematic Natures and Philosophical Questions
In amongst the usual obfuscatory dross of yet another post desperately trying to explain the arbitrariness of philosophy in a way that makes it sound even slightly less arbitrary, I read this gem.
Questions make other questions disappear. How so? By disappear, you mean no longer require answering, or no longer interesting? Could you provide an example of a question that has 'disappeared' together with the question which dispatched it with such lethal finality?
I think it's a pretty banal point. Whether something seems like a relevant philosophical question depends a lot on from what standpoint you're doing philosophy. Have a short list of examples.
Wittgenstein thought he dissolved all (or most) philosophical problems with his Tractatus.
Husserl thought he refuted the need to answer the skeptic with his phenomenological reduction.
Heidegger thought he dissolved Cartesian views on mind and even subjects as traditionally understood.
Strawson accuses Dennet of spouting 'learned nonsense'.
Dialethic logic as Priest presents it makes a lot less sense if you take Prior's approach to the Liar.
Stove's refutation of 'worst argument in the world' alleges to refute all idealism and contained problematics.
Laruelle plays trumps with most of philosophy saying it can't think of the real without reducing it to a philosophical abstraction.
All of these people have some idea of what it means to do philosophy; what it means to address and formulate philosophical problems; and whether some of those formulations are even possible, necessarily false or even nonsense masquerading as sense. It's a very, very common 'move' to reframe most of philosophy in the way you do philosophy.
Maybe a more interesting question: do you think philosophy should be describing how natural processes work in general? Or does it merely use descriptions of natural processes to facilitate interpretations? And is the philosophical discourse circumscribed by those kind of analyses/interpretations metaphysics?
If you think it's within the ambit of philosophy to do the former, there's a lot of people that disagree with you. If you don't think the former's it's within the ambit of philosophy, there's a lot of people that disagree with you.
If you think either are relevant, you're in disagreement with Heidegger's idea of metaphysics. If you do metaphysics in broadly the same vein as Heidegger but use natural processes to inform your metaphysics, you're doing a Merleau Ponty and subverting Heidegger's methodology in an interesting way.
None of this is arbitrary, it's all well motivated theory. No matter where you situate yourself you'll draw boundaries of relevance somewhere. The position where all philosophy seems arbitrary and pointless is still a metaphilosophical position, and as the former post illustrates, to do philosophy is to do metaphilosophy; and it should be obvious that metaphilosophy is philosophical, right?
To your first post - You've provided me with a lot of Philosophers who thought they'd made previous philosophical questions redundant. I'm unsurprised that you'd think me so entirely inane that I'd be unaware the some philosophers thought they cracked something. What I was asking about was how the 'questions' dissolved other questions, not the answers. What you've provided is a list of philosophers who thought that the 'answers' to their initial questions dissolved other questions. This phenomenon then seems simply to be a feature of inquiry, not philosophy. If I decide that god doesn't exist, the question 'What colour is god's beard' obviously becomes redundant. This is no different to Physics where the discovery of photons made all questions about the nature of ether redundant. So I'm not seeing the meaning behind what you wrote yet.
Your second post is much more interesting, you say "arbitrary and pointless" when I only said "arbitrary", why the additional term? Why is arbitrary automatically assumed to also mean pointless?
The fact that meta philosophy is also doing philosophy is exactly what I'm talking about. The way in which so many posts are written explaining "the way philosophy is", when that "way" is invariably something ambiguous like 'framing' or 'asking the right question' or some such.
Philosophy seems pinned between two attacks, those who demand it reveal what 'truths' it has discovered or else be consigned to the rubbish heap, and those who claim the whole thing is nothing more than a series of works of art, you either like or you don't.
What intrigues me is the convoluted arguments used to get out from this pincer manoeuvre. Of which I should add, this thread is the latest example. It amounts to "OK, I admit philosophy does make arbitrary choices, but look so does maths so its not that bad after all"
I mistook your intentions as a usual 'what's the point in philosophy anyway' poster, albeit an articulate one.
I think you're right that it's a feature of inquiry. But it's also a feature of worldview, perspective and personality. I'm sure you'll have had arguments with people, say housemates or partners, where one of you did something completely usual and inconsequential for you but to them it's a big deal - a gripe or a blessing. The same can be said for problems in philosophy; though, problems in philosophy motivate more philosophy; some of which is developing new problematics. Which is probably related to how the philosopher (and philosophy) looks out on the world and what problems they're motivated by. Sometimes it's intimately personal and political like Bell Hooks, sometimes it's abstract generality like Kant.
I think you gave the conditions under which this is a non-problem in your post, ironically. All inquiry functions like that, thinking critically, rationally and creatively are all part of inquiry in general - they don't suddenly disappear or lose their general character when the inquiry is philosophical in nature.
The thing to note here is that it's pretty rare that a philosophical problem could be called 'purely philosophical'; philosophers typically care about a definite space of problems. Witty cared about doing right by language, Heidegger cared about doing right by being and subjectivity, the positivists cared about grounding science in a scientifically flavoured philosophy; doing right by science. So did Lakatos, Kuhn and Popper in related but contrary ways. Foucalt held science in something like an anthropological epoche and cared about it as a discursive practice. They all cared about something, and they inquired in the way people inquire; inventing along the way and following their noses along historically conditioned trails of expressions (islands of sense in the previous post) to do it.
I also read you as dismissing what I had to say because the vocabulary in my responses to @csalisbury was unusual. We usually have a similar perspective on things and similar philosophical tastes, so I can use a playful shorthand with him.
I'm not sure I can agree with this. All inquiry certainly dissolves previous questions when a new path is chosen (very much the way maths is described here), but what separates philosophy, and singles it out for this kind of attack, is that it vacillates constantly between the two ideals seemingly in response to the attack itself. Heideggar didn't think he was just offering a work of art to world, to be hung on the wall of those that found it appealing but justly rejected by any for whom it wasn't to their taste. He thought (and wrote extensively in his notes) that he was actually discovering something real.
The thing about maths (in common with other disciplines) is that it is circumscribed entirely by the 'choices' that have been made. If I were to answer a complex equation on the presumption that irrational numbers did not exist, I would simply no longer be doing maths. If said that the answer to 2+2 is 5, because 5 looks nice (having rotational symmetry with 2), it would not be the case that I was wrong, everything in my statement is logical and true, it would be that I was no longer doing maths.
Physics has the same definitional constraint, I might describe the nature of light in terms of God's message to mankind after the great flood. Again, I would not be wrong, just no longer doing Physics.
Philosophy has no such exclusory definition. There's nothing that definatly isn't philosophy, other than it being one of the other disciplines (physics, maths), it's more like the arts in that respect (the perennial 'what is art?' question sounds suspiciously similar).
None of this is a problem in itself, I don't think. Art seems to get by quite satisfactorily without anyone having yet answered its foundational riddle. It becomes a problem for me when the ambiguity about definition is used to shut down lines of enquiry others are finding useful. Too often I hear "Logical Positivism has been disproven", "Kant showed that...", "[X, y or z] is not even proper philosophy", "you can't comment on X until you've read y". None of this has any justification without a definition.
I'm not interested in defining philosophy when the definition of philosophy would itself be a philosophical problem. Inquiry proceeds without a satisfactory definition of it, so do all of its manifestations. I'd be as well wondering how you could post without being able to give a necessary and sufficient condition for a given object to be 'part of a language'.
Even math has somewhat ambiguous boundaries - at what point does it become logic? Is set theory logic, like it is historically, or is it part of mathematics? Is it as some mathematicians treat it like 'metamathematics' - then what about model theory, is that mathematics despite sometimes being called metalogic? Where does pure mathematics end and applied mathematics begin? When does statistical physics collapse into statistics or applied statistics become statistical physics?
Those questions need answers, the continuation of every single discipline requires immediate clarification of what they're doing!
I think all of these are largely pointless questions. If you insist that something have a clear definition before we can begin talking about it, or even begin to circumscribe its sense (how are definitions made without this stage of prefiguration?), I've got no interest in continuing the discussion.
The prejudice in your first presumption (that by arbitrary, I must mean arbitrary and pointless) is clouding your interpretation of what I'm saying. Unless my exposition is considerably more suggestive than it reads back to me, I don't see anywhere in which I state, or imply, that philosophy 'must' do anything at all, merely that it hasn't, and that this absence has implications.
In fact, I'm struggling to see how anyone could read my post as saying anything except the exact opposite. That philosophy most certainly must not try to define itself so strictly because to do so just leaves behind a whole raft of thinking which then has to label itself. Philosophy is, by necessity, everything that isn't something else.
Quoting apokrisis
But this isn't the goal at all. The goal is to respond to problems as they arise, and forge the necessary concepts to deal with them in situ. Taxonomy ('the division of reality'....) always comes after the fact - not too far removed from taxidermy. 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.
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.
I'm not going to assume the silly Socratic game of asserting power by questioning you on what you mean by terms, I'm responding in the way that I am precisely in order to try and convince you that it really doesn't matter; that is, it has no meaning for philosophy, to have a sufficient and necessary condition for what it is. Or even a more lax dictionary definition (they exist, of course).
Google/Webster gives it as:
I'm reading you as equating lack of definiteness in specification and lack of any substantive characteristics. This is suggested in:
I'm pretty sure that those shut down attempts are part of the positive character of philosophy. Like what distinguishes it from rhetoric; philosophers are supposed to care about the true nature of things. A speculative realist and a correlationist are going to be at odds methodologically; some threads of ideas will see problems in other threads of ideas which aren't native to that thread of ideas... And characterising what is and isn't native (characteristic, necessary for, required by, condition for the possibility of, presupposing the material conditions of...) to a set of ideas often in novel ways is absolutely part of the analytic stock and trade of philosophy.
J.L. Austin:
Heidegger:
and it is a thoroughly excellent part.
Oh, also; I discovered a food allergy last night, got one hour's sleep and I'm running out of pile cream. I'm probably not in the best place for understanding detailed prose at the minute.
But in your previous comment you asserted that defining philosophy was itself an act of philosophy. If so, how could the absence of an agreed definition possibly have no implications for philosophy? It's opened an entire line of investigation for a start, as you yourself have suggested. Not to mention the fact that it allows Derrida and Russell to be considered as part of the same subject. Can you otherwise define what connects 'Glas' to 'Principia Matematica'?
I would fix the bolding part: it'd be a case of giving up the idea that there are irrational numbers by denying that such areas can be measured ('are amenable to measurement at all'). It's actually worth quoting Heller-Roazen in full on this point:
"The Pythagoreans, however, were no strangers to the uncountable. Although they barred numberless relations from the domain of their arithmetic, they also named them in no uncertain terms. They called them “unspeakable" (?????o?), “irrational" (??o?o?), and “incommensurable" (????????o?). From such appellations, one might infer close acquaintanceship. Yet the familiarity the classical theorists of number possessed with such relations could not be knowledge, according to any classical standards of science. Infinitely eluding the rule of unity, incommensurable quantities could not be considered to number anything that was and that remained a single thing; for this reason, they could hardly be considered to number anything at all.
Of such unspeakable relations, it could only be deduced that, like the impossible root of the single tone, they could be no collections of one. They were, quite simply, immeasurable, and as long as every definition in arithmetic and music was to be numerical and every number was to be discrete, they were unrepresentable as such. They might well have been somehow manifest to the Pythagoreans, but, being uncountable, they could be no “remainders.” Their sole place was at the limits of their art of quantity".
--
It's also worth noting that our conversation so far is almost like a case study in what it means for how different categorizations commit one to different parsing-out of concepts: "it's not there there can't be squares like that; it's that they can't be measured"; In some sense, this is a 'choice' too: perhaps one can indeed deny that there are squares covering an area of two square units; but one would have to make the corresponding move of then saying something like: 'the things you thought were squares covering two square units are not squares; they are ?quares'. Wittgenstein's 'rule-following paradox' was all about this: that every move in a game can be said to accord to a previously undiscovered rule, without breaking old ones. But these new rules are not just shuffling of goal-posts: they make one see things anew - if done right.
So there's a kaleidoscopic or rubik's cube aspect to philosophizing: twisting a knob - a concept - in one way, ramifies throughout the whole series of 'possible' concepts and implications.
I'm sorry to hear that. I will await any response (or not), as and when you feel so inclined.
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:
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).
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.
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.
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.
Quoting 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.
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:
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?
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?
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, 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.
[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]
Quoting 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.
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?
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.
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.
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.
Quoting 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).
Quoting 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?
Quoting 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:
Quoting 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.
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.
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.
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.
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.
Quoting 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.
In case I left you confused - it does happen - I hope it is clear that symmetry-breaking is what connects a triadic system of symmetries. So the "ultimate symmetry" would be a three cornered structure, if you like.
You have the symmetry of vagueness - a state where (material) contingency and (formal) necessity are differences not making any difference. As Peirce noted, the PNC does not apply.
You have the symmetry of generality - a state where globally there is the continuity that has formally absorbed all possible differences so that they don't make a difference. As Peirce noted, the LEM does not apply.
Then you have that final symmetry of atomistic particularity. Eventually, even constraint no longer makes a difference. Locally, things arrive at the ultimate simplicity of a geometric point, a mathematical identity element, a quantum particle, an informational bit, a semiotic mark, or a fundamental entropic degree of freedom.
Or as the laws of thought would have it, the emergent entity to which the principle of indiscernibles does finally successfully apply.
So symmetry is something to be understood in a formally general fashion - hence why maths is the domain that winds up speaking about it.
But in a holistic metaphysics with a triadic structure, we are talking about three kinds of symmetry-producing limitations. If we pursue symmetry-breaking back to its source, we find it in three types of bounds - Peirce's triad of firstness, secondness and thirdness. Each is a "level" of symmetry - a terminus to a dichotomous "other".
Kind of you to say so, and while it’s true I have an interest in Buddhist philosophy and also philosophy of math, I skipped undergrad Formal Logic so I don’t think I really have the chops. I will have to content myself with the occasional parenthetical comment. [By way of consolation, here is a podcast I listened to recently - an interview with Graham Priest on Buddhist Logic.]
This appears to be the same point I have made at various times with that silly philosophical game in which players make up the rules of the game as they go along.
So, Cheers!
Yet, that’d work as prescribed. It would serve the purpose of anti-metaphysicians. ;)
But this here is the very move that is unmotivated: it responds to no imperative other than your 'will-to-system', which, as Nietzsche rightly observed, simply lacks integrity. It is an intra-systemic imposition that responds to no genuine, worldly problematic; it's less the revelation of a 'basic structural mechanism at the heart of existence' than a transcendent, theological principle posited from above. It breaks with the demand for immanence, and, like I said, confuses description for prescription.
Heh, kind of. But that game was too arbitrary: it wasn't made for a purpose. The distinctions articulated within it were not posed to solve anything in particular. It's closer to say, what Apo generally attempts to do than what I'm trying to do here.
Maths deals with symmetries in Group theory, and those mathematical tools are used by physicists and other scientists to model reality and this or that part of reality. Does that tell us anything about reality, or does it just tell us about the way we currently model that reality?
Sounds legit.
That was part of it's attraction for me; it describes wheels spinning without ever engaging, an aspect of much of philosophy. It's, as you set out, the use to which we put the rules that makes for one choice over another.
Fair enough. I suppose it's that very distinction, between the engagement and not, that interests me. Some here seem to think that philosophy per se is wheels in the void; I want to defend its friction - while avoiding at the same time a certain positivism, ugly and moribund.
... or gravitons aren't even part of the Standard Model yet.
Quoting jkg20
I'm not asking you to deny that the Standard Model is "only a model". The clue on that score is probably in the name.
The issue here is SX calling particle physics use of symmetry breaking "arbitrary".
So do you think group theory is arbitrary? Are its results contingent in some fashion you can explain?
And is particle physics success in using symmetry maths to account for particle relations arbitrary? When a model fits like a glove, why would we have reason to think that was also merely contingent?
Of course, it could be a lucky accident. No one can deny Descartes his demon.
Your illiteracy has reached new heights I wasn't sure possible.
Really? Because it sounds remarkably like you want to defend your favourite kind of philosophy against charges of obfuscatory meaninglessness but reserve your right to dismiss any philosophical positions you don't like on exactly the same grounds.
How do we account for the usefulness of pure mathematics in describing and predicting reality? That's a different question, but I'm certainly not convinced that the answer to it requires either mathematical realism or physical realism.
Sure. If you read my posts, you will see I am a pragmatist. You’re talking about bread and butter epistemic issues.
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?
See, this too I think is just wrong. That is, any close attention to philosophy shows it to be an incredibly constrained practice. I quoted Deleuze in a previous thread as saying that philosophy is nothing but the pursuit of the implications of a problem posed; a 'great philosopher' being one who is equal to the task of that pursuit, doggedly following it wherever it goes. I mean, everything in Plato unfolds according to a certain committent to Reality of Ideas (?????) and their status as surpa-sensible; everything in Leibniz unfolds according to a commitment to a conception of the principle of sufficient reason; everything in Bergson follows from a commitment to the primacy of duration, etc.
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.
Might be worth mentioning here that while I despise say, Plato and almost everything he said, I accept that his project was a rigorous, well-drawn one for all that; one doesn't have to like a philosophy to recognize and respect its consistency and power. In fact I find him all the more dangerous for it.
You're mistaking the boundaries set within schools by the restriction of their premises with the boundaries set within entire disciplines on the "schools" that can be contained therein. It's not that, once a commitment is made, philosophy is still less contained than maths, it's that commitments of almost any sort can be made (and followed) and still constitute 'philosophy'. This is not the case with maths, where commitments of the sort you describe are still made and followed, but the range of possible commitments is severely constrained by the nature of the broader investigation (the whole of maths, sensu lato)
But of course it's the case with math. The history of math is nothing other than the history of commitments made one way - and not another. And even this too is unfair, because plenty of mathematicians do explore those paths less travelled as well. The only thing to add to all this is the pragmatics: what motivates the commitments. But this is not something that can be given a priori, nor argued to be the unfolding of some divine plan a la our resident theologian Apo.
I haven't said maths doesn't have a variety of commitments, some of which are "less well-travelled", in fact I said the opposite. The point I'm making is that it is not sufficient to support your argument simply by stating that maths has a variety of commitments whose properties then constrain further investigation, and so does philosophy, so they're basically the same.
What differentiates them, I would argue, are the types of constraints that are placed on those commitments, the breadth of the field. It is easy to describe what isn't maths by reference to the activity alone. I think it's considerably harder to describe what isn't philosophy, without reference to other subjects. Even harder to describe what never will be philosophy and hope to have any certainty that you'll be correct.
The point is that the reason why philosophy is a wider field is to do with significant, categorical (non-scalar) differences in its approach to concept determination. Specifically, it's criteria for both the selection of commitments it is interesting to follow and its criteria for testing the degree to which they have been followed. The difference being that maths has reasonably strict criteria for both, where philosophy has virtually none.
Give any work of maths to an educated layman. What are the chances they'll correctly identify it as maths? Give a potential work of philosophy to a group of philosophy professors and even they won't agree on whether it is one or not.
This is a comforting opinion to hold I guess, but as it stands there's nothing here but assertion and some cute imaginative scenario posing.
*waves hands in the other direction*
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.
Quoting schopenhauer1
I don't think you've properly elucidated the pertinent parameters. You've defined consensus in terms of constraint. However, consensus is properly defined in terms of agreement. And there are two principle parameters to agreement, 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".
Quoting schopenhauer1
In order to even count something intelligibly, a consensus on what constitutes a unit is required. If "a unit" is understood as an individual being, then a treatise on being, and consensus on "being" is required before we can have an accurate count. 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.
Quoting schopenhauer1
I think that this is exactly right, but you misconstrue the implications, the conclusions to be drawn from this situation. 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".
Quoting schopenhauer1
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.
Quoting apokrisis
No, no, this definition of "unit" must be rejected as circular, or an infinite regress, and therefore not a definition at all. If a symmetry is a unit, and we refer to a prior symmetry-breaking to define that unit, then we have defined that specific unit, by allowing that there was another unit, and referring to this other unit, which is prior to that unit. You have defined "unit" by referring to the unit which birthed it. Therefore there is no definition of "unit" here, just a circle a circle, defining unit with unit, which if pursued would lead to an infinite regress of units. Such a definition is inherently contradictory, because the unit is referred to as a symmetry, but you move to define it by referring to a symmetry-breaking. Therefore what is defined is not the unit, (the symmetry) but the annihilation of the unit (symmetry-breaking).
Quoting apokrisis
As I said, mathematics could be based in "zero", but this entails a negation of the concept of "unit". When the unit is eliminated, annihilated by symmetry-breaking as you suggest, to base the mathematics in zero, then we need a clear and precise definition of what "zero" refers to. We nave no more "units" to base this mathematics in, only the assumption of "zero". Without a clearly defined "zero", all this mathematics employs completely arbitrary zeroes (concepts of zero), and is just random nonsense.
Quoting apokrisis
Your global mathematics here is based in "zero", symmetry-breaking, the annihilation of the unit. You also refer to a local mathematics based in units. Instead of attempting to work out this incompatibility, by establishing a properly defined relationship between zero and the unit, you simply introduce contradiction into these two with the necessary implication that symmetry breaking, annihilation of the unit, is derived from symmetry, the unit, and the constraints which constitute local unit are somehow derived from local freedom, lack of constraints.
Quoting apokrisis
It is not mathematical physics which tells us this, it is a simple ontological assumption. As I explained to Schop1 above, the mathematics follows from the assumptions. There is no inherent need to describe existence in terms of broken symmetry, that's a choice.
Well, I am a layman when it comes to maths and have never yet failed to identify it, and I've been in a room full of philosophy professors who couldn't even agree whether a work was 'philosophy' or not, so it's not entirely imaginative scenario posing. 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.
Then you didn't see when I said:
Quoting 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.
On hand-waiving:
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.
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.
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?
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.
Quoting 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.).
Quoting Metaphysician Undercover
Yes, but I think there is more consensus in math based on the more limited scope of math.
Quoting 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.
Quoting 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.
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.
Great.
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.
The whole post was good, really good, but this is my favorite part.
Carry on.
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.
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.
You're welcome?!
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.
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 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.
No, wrong. Explained already.
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.
Quoting 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.
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.
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.
...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...
PoMo is going to get such a shock when it catches up to 1980s work on universality in dynamical systems. :)
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."
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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."
So this means you think math dictates the uses to which math is put? Is there a theorem for that?
Quit complaining and put up a counter argument if you have something to say.
I really don't understand what your project is here. It just comes across as someone labelling all the parts of a car down to the last bolt but refusing to actually drive it anywhere.
If you're interested in problems within a post-structuralist framework there are a large number of really interesting ones to tackle (so I'm told), but what you seem to be doing here (and others too, I'm not singling you out specifically), is expecting answers from within a broadly post-structuralist framework to apply to the structures which themselves contain that framework. There's a reason why academic philosophers work, and publish, almost exclusively on problems set by their chosen field (ethics being a typical exception). The analysis simply doesn't make sense outside of that field until it is 'translated' into whatever language the reader/critic speaks.
All that's happening in this whole thread, and dozens of others like it on this forum, is that ideas are presented within one framework which are, by and large, fairly routine (not uninteresting, just sequential) and they will be met by others from within that framework by an overabundance of cross-referencing proofs, or by others outside of that framework trying to do the 'translation'. That's what @schopenhauer1 is trying to do here (and @csalisbury, to an extent), but instead of helping with that translation, you're just responding like the clichéd English tourist refusing to speak the native language. You're saying the same thing louder and slower in the hope they get it, in your language.
I can really only see three reasons why anyone would want to post ideas on a general philosophy forum. Either they carry such inductive weight that they are virtually impossible to deny and so are useful out of the box (... is that a flying pig?), or they're promising but unfinished and could benefit from critique within the framework of the problem they address (in which case, if you don't clearly specify that you're going to get nowhere), or they're ready for others to 'try on' to see how they work, like taking the car for a test drive.
With obviously varying degrees of skill, that's all people here are trying to do, test-drive the ideas presented, within their framework, the fact that their feedback isn't then going to be in terms of the problem the post set out to solve is not a flaw, its inevitable. If an idea is going to be of any use to anyone (surely the only reason for posting it) then it's going to solve a problem they have, which is going to take quite a bit of collaborative translation.
Hey great post.. One of the best arguments on this forum for the principle of charity and collaboration in general. That should be in a guideline or something :up: .
I'd actually like to thank you for taking the time to construct this post.
Quoting StreetlightX
I get it- the creativity before any formal proof is written. But I am trying to say that the creativity is still dictated by the limited scope and content of math itself contra philosophy where the game is much wider. The constraints are as limited as you want to make it in philosophy.. The constraints are very tangible in math and dictate certain outcomes and legitimate moves which are even possible within its confined scope. This constraint in creativity, in a way, creates a confined area that I am claiming is more deterministic (the outcome shall reveal itself). The creativity is not nearly as boundless as other subjects, thus the creativity is more dictated by the game at hand. The proof after this constrained creativity, is just more constraint- this time on what a legitimate explanation (in the community) can be- another difference but at a more concrete level.
Sure, and I acknowledged this: but the relevant question is what accounts for this difference in scope. The obvious answer seems to be that math is constrained by its subject matter: it deals, roughly - very roughly - with with numbers and their operators. Philosophy is obviously subject to no such restriction; or at least, it is subject to restrictions of a different kind. But regardless of their respective fields of inspiration - one narrow, one broad - the point is simply that the dialectics of necessity and contingency function in the same way. Again, I think you're massively underplaying the way in which, once a philosophical problematic is set out - a concept developed, a problem articulated - the moves are just as constrained as they are in math.
Thanks, glad it made sense to someone at least.
Quoting StreetlightX
But this is the game, do you not recognise it?
Neither can a point have an edge, nor can a circle be made up of straight lines. So this idea is contradictory in two ways.
Quoting apokrisis
A point marks the limit to a line segment. It is contradictory to say that a point is a line segment which can't be cut any shorter, because a point and a line segment are fundamentally different. The two are incompatible. A point has zero dimensions, while a line signifies a dimension. A point has absolutely no spatial extension. It's relation to space is limited in a most complete way such that it cannot have any shape or spatial form. A line, despite the fact that it continues infinitely, has a very specific, and limited spatial extension, limited to what we call a dimension.
Quoting apokrisis
I agree that a plane is two dimensional. But a line, by definition cannot be two dimensional. Therefore, "curved line" is itself a contradiction. To express two dimensions with lines requires two distinct lines. The relationship between the lines is expressed as an angle. You cannot have a curved line. That's why pi is irrational, it tries to establish a curved line, but a curved line itself is contradictory, irrational.
My argument is that it is not the curved line itself which produces the irrationality, it is the relationship between two dimensions which is what is truly incommensurate, just like the relationship between zero dimensions and one dimension, described above, is incommensurate. That is why the square root of two is irrational as well. What this indicates is that our spatial concepts, in terms of dimensions, are incorrect. The concept of dimensions of space produce an unintelligibility and therefore must be incorrect.
Quoting apokrisis
I think that the application of the theory of general relativity has proven this to be false, the shortest distance to connect two points is not actually a straight line. This is further evidence, that our dimensional modeling of space employing lines and angles, vectors, is incorrect.
Quoting apokrisis
I answer this question by saying that the entire conceptual structure which models space in terms of distinct dimensions is inadequate and therefore incorrect. This conceptual structure leaves us with an unintelligible, irrational relationship between dimensions. The relationship is modeled with angles, but the concept of "angle" doesn't allow for the true nature of curvature. The "angle" is something totally arbitrary, inserted into spatial conceptions as an attempt to alleviate the described problem of an incompatibility between linear dimensions. When something fails you insert a stopgap to deal with the problem. That is the "angle", but the stopgap is supposed to be temporary. The only real thing that the "angle" represents is the limitations of linear geometry. Pythagoras was perplexed, that there was irrationality inherent within "the right angle", as right and useful as it had proven to be. In reality, as useful as it may have proven to be, no angle is the right angle because "the angle" represents nothing, it is a falsity.
I'm interested in how you're thinking of 'accuracy' of an idea. If you apply it to the steps of proving a theorem, that's usually pretty banal; people who can read the language the proof's written in will usually be able to see if it's right or not, if the general idea is right etc. There's a derived sense in which a theorem can be accurate; if its proof is. But I don't see that this sense of accuracy applies to mathematical objects or ways of thinking about mathematics in the same way.
There are probably, to borrow from Austin, felicities and infelicities in how theorems are proved and how people think about/imagine mathematical objects. I tried to highlight this with the Dedekind Cuts vs Cauchy sequences thing. Street's article highlights it in a more accurate way; a particularly felicitous way of thinking about the problem lead readily to its solution.
An apocryphal quote (might be real, couldn't find a reference when searching though) attributed to Grothendeick is that 'don't try to prove something until it is obvious'. Similar one by Riemann 'if you give me the theorem I'll give you the proof'. Mathematicians seem to think about creativity in mathematics this way; a certain 'accuracy of ideas' which doesn't immediately reduce to the accuracy of a proof.
Grothendeick really, really thought of stuff this way:
You said you're unfamiliar with math. An un set up problemscape for you might be solving this integral:
[math]\int\limits_{-\infty}^{\infty} e^{\frac{x^2}{2}} dx[/math]
and I can give you a couple of hints to its usual solution:
I think if I gave that integral and the hints to some math undergrads with university level calculus experience on a homework assignment, a good chunk of them would be able to solve it. In less than an hour. With no Wolfram Alpha. An un-setup problemspace, to you, might be me just giving you that integral to compute with no hints. It has a certain impenetrability, you need to spend some time on the contours of the problem before finding a way in. When you're about the solve the problem, if you're anything like me, you'll be like 'oh god... I can do this*? This will actually work? Holy shit that's cool'. By * I mean 'I can use some details of bullet points 2 and 3 to solve it? Wow.'.
Of course, you could google that integral if you know how to use Google to search equations (standard internet equation notation), then you'd find the standard solution to it. You'll discover that the problemscape for that problem is very well studied, and there are lots of avenues of study leading to and from it.
I imagine whenever you post something on here, you have an opinion on the topic. If the topic's something you've studied before and you're regurgitating pre-articulated thoughts it doesn't require much thinking except in how to communicate what you've already thought. You know your way around the problemscape; the imaginative background you have for the problem; and all that remains is to beam it into your readers' minds with sufficiently good writing.
I don't think you're doing that here, you seem to be in a similar discursive position to me, you've got some intuition on the structure of philosophical problems in general, you see what I'm saying as a flawed instance of that structure (at least on the forum), and you're struggling to find the words to articulate in response to what I'm saying to communicate a disagreement or difference in emphasis. [You've got to wrap up what you think into a neat package; which perhaps can be identified with thinking about the problems in the thread and in your head; then translate the package into a common language.] The stuff in the [] seems to occur all at once.
If you relax the assumption that we're 'beaming things into eachothers' heads' with language use, and instead inhabiting a vast terrain of communal ideas, we get a more interesting location for the problemscape. To be sure, it's something we're doing, but it's also just as much something that's already been touched on by countless others in different contexts. Sure, what we're doing isn't going to be inscribed in the canon of philosophical discourse since we're on a forum, but that doesn't matter that much. It's like we're both on a knitting forum and sharing patterns.
So, I resist your claim what we're doing is mostly driven by psychological peculiarities - we're navigating some abstract space of problems that we share but have our own different copies of. I see the claim that we're driven by psychological peculiarities as misguided: it's like saying the knitter is driven by psychological peculiarities; sure, they are in a sense, but they're also following a pattern or making one up.
I'd like it if one of our few academic philosophers could chime in on their experiences of dealing with philosophical problems. @Pierre-Normand, do you have any input on what it feels like and how you address philosophical problems as an actual academic philosopher?
Ok, so we are essentially accusing each other of the opposite thing- you think I am making philosophy too broad and overlooking its constraints, and I think you are making math too broad and under-emphasizing its constraints (and the outcomes there of).
This may go back to something you said in the OP about the fundamental metaphysics/epistemology of math- is it invented or discovered? I haven't thought about it much, but whether numbers themselves are invented or discovered, the logic/processes/patterns involving them seem to have a "discovered" aspect to them. The limited rules dictates next moves more than it is simply synthesized by the "limitless" inspiration of the mathematician. In other words, if this game was run again, roughly the same models and demonstrations would be constructed. It would not be perfectly the same, line by line (there is some contingent variation in symbolic communication), but on the whole, the concepts behind these mathematical proofs would shake out. Perhaps in a particular framework, philosophy can work similarly, but other camps would negate this process. So initial creativity is there for both, but beyond that, the way it works out (the discovered-like process of math/ consensus vs. the more synthetic-like aspect of philosophical moves). But now we are getting into the epistemic differences between the two.
Edit: Granted, there are some conventions (like certain statistical constants, etc.) but this is recognized as such in the name of pragmatic reasoning.. which this field is especially amenable to. I know apo mentioned using 360 degrees being contingent, but again, the "discovered" aspect I refer to are the concepts behind them.
So a point can be the edge to a line? Make up your mind.
Quoting Metaphysician Undercover
So if we cut away all the line to one side, it is bounded by a point on that edge. And if we then cut away all the rest of the line to the other side, what then? Is the point bounded by a point or is there just the point?
Isn't the fundamental difference that the point is the natural unit of which lines are composed? It has a locally emergent symmetry which marks the place where no more can be cut away. A point is all featureless edge?
Quoting Metaphysician Undercover
But doesn't the point have a location? It exists as a limit on dimension. A line represents a space of points. A point represents that space without a line.
Quoting Metaphysician Undercover
Seems radical. But sure, throw away the concept if you can tell us about something better to replace it.
Or maybe it is your logic that lets you down. My logic expects metaphysical truth to be rooted in the fruitfulness of dichotomies. Yours is instead deeply troubled by discovering dialectical "contradiction" at the heart of things.
Quoting Metaphysician Undercover
You got it exactly backwards. I was saying GR indeed shows that "straightness" is relative. It cashes out as the shortest path, the least action principle, once your geometric intuitions include some proper notion of action or energy density along with the spacetime dimensionality.
That is one of the issues with classical geometry. It all rests on spatial intuitions. It doesn't have a natural way of including energy, thus actual time and change, in the picture. It is a story of the directions but not the actions.
So GR is geometry with energy density included. It is more physically realistic as maths. And now the least action principle comes to the fore when we are thinking about "straight lines". We have generalised Euclidean geometry so that there is just universal curvature. We have removed a major constraint - the one picked out by the parallel lines axiom. And now straightness becomes defined more clearly as the shortest possible (energetic) distance between two points.
GR changed the simple definition of what counts as straight. That was the bleeding point.
Quoting Metaphysician Undercover
Seems coherent - in being dichotomistic - to me. Dimensions are defined by being orthogonal to each other. Mutually exclusive and jointly exhaustive. The x axis is oriented so that no part of it - apart from a point, an origin - intersects with the world of the y axis.
You can head for infinity in the x-axis without moving an infinitesimal amount in the y-axis. And that is your strong definition of a spatial dimension. You have two disconnected or asymmetric directions that exist for motion~rest.
Remember how that distinction played out in Newtonian mechanics. Spacetime became defined by its energy conserving transformations. Masses could spin inertially on the spot or move inertially in a straight line. These were local symmetries that couldn't be broken down as they were the terminus to the very possibility of a more global symmetry breaking.
Then along comes GR, and even QM, to add the missing ingredients of time and energy back into this mathematical picture. The classical view was certainly correct, but still carried extra constraints that proved themselves to be local and particular rather than cosmically general.
Quoting Metaphysician Undercover
It is not arbitrary. What got inserted was the very notion of a dichotomy or asymmetry. Dimensions are distinct due to their orthogonality.
Ask yourself why pi = 180 degrees. Hint: a circular rotation that flips you back to a flat line having transversed its orthogonal "other".
These things aren't arbitrary at all. They are logically founded. They are Platonic strength.
That is why SX is so off the mark in his OP. Maths is "unreasonably effective" because - where it is based on the metaphysics of symmetry and symmetry-breaking - it is finding ways to model the necessary structure of existence. For anything to be, it would have to have being in this self-justifying form.
That was MU in fact. I agree with you that the fundamental structures of mathematical thought are the inevitable rules of form or constraint that are there (so "somewhere" a bit Platonic) to be discovered.
So where does the arbitrary or the contingent come in? Two ways.
First - the Peircean point - the deep structures or constraining rules must be themselves emergent from a ground of arbitrariness. They are precisely the states of organisation that will emerge from chaotic possibility itself. Order must evolve to regulate instability. It might be a very minimal order, a very permissive order. Yet still it will be a generalised state of order.
Even statistics depends on bounded action. Randomness can have macro properties like a mean or a variance because there is some kind of global constraint bounding a system of independent variables. You get a temperature or a pressure only when your gas is confined in a flask. And any workable notion of randomness or probability depends on a duality of free local action coupled to definite boundary constraints. Otherwise there just wouldn't be any "statistics" - any macro properties to speak of.
So our very notion of the arbitrary or the contingent only makes metaphysical sense in the context of its "other" - the necessity, the regulation, to be found in some set of bounding constraints. You can't even have the one without the other. Hence there is the Platonic structure to be discovered as the necessary spine of existence. That can't not be the case ... if you do in fact believe in the matching "other" of the accidental or contingent. Each secures the reality of the other in complementary fashion. Hence why SX's orientation, as expressed in the OP, is so off-base from the start.
Then second, in discovering the deep and necessary structure of existence, we humans can gain our own local possibilities of control. We can insert ourselves as agents into the cosmic equation. If we construct mathematical models that encode the basic rules of the game - the game that is symmetry and symmetry breaking - then we can start to use them in our own "arbitrary and contingent" fashion.
We can do things that Nature doesn't seem to be thinking about or caring about. And folk of course find that a big philosophical deal. Suddenly our values and desires start to affect the metaphysical story. We are the source of meanings. We are the source of inventions. We are the source of fictions and fables and beliefs.
All this can lead to a PoMo-style rejection of metaphysical structure and meaning. Dichotomies and hierarchies become dirty words. Humans can transcend nature to become ... well, now it seems philosophy has solid grounds for its Romantic revolt against all "merely" physicalist constraints or necessities.
But this gets the metaphysics wrong. Constraints only constrain. So there is room enough in our cosmos for the very mild and attenuated constraints - the generalised thermodynamic tendencies - and then the much more complex and particular kinds of natural organisation that are reflected in evolved life and mind.
There is no basic conflict here, no real battle that the Romanticist must fight. We don't need to be like SX and fetishise the arbitrary and the plural, anathemise the Platonic and the unitary.
To get back to the 360 circle, I would note how the choice of 360 wasn't so accidental. It seemed important that we find a numbering system that made division into simple fractions easy, while also offering enough divisions to capture the differences that were of (Babylonian/astronomical) interest to us.
Again, some deeper symmetry was the reason for 360 being a choice that lasted. It was a number that offered the kind of symmetry breaking we found most convenient. Given an infinity of numbers we could have chosen, picking on 360 was not an arbitrary act from the point of view of a human having to do the calculations on a regular basis.
So inference to the best explanation - the principle of least action in practice. We jiggle the bits about until it all snaps into place with a holistic best fit.
That is, we start with a broken symmetry - some "problematic" that is a collection of disjointed parts. And then we probe for the symmetry, the global coherence, that must have originally connected them.
Proof follows because that is the formal (re)construction. It is the creation of the bottom-up deductive path that connects us securely to the top-down glimpse of the Platonic reality.
So first comes the abductive leap that allows us to see the fragments in an inductively retrospective light. We see the smashed glass across the floor, the cat innocently licking its fur on the bench. In a flash we see how the symmetry of the vase got broken.
Then proof is the mopping up operation. All the parts get gummed together to show that the vase did exist as we imagined. We have sound reason to blame the cat as we are certain the vase wasn't just stuck in a cupboard while we weren't looking and meanwhile some random collection of glass just materialised on our floor.
Somewhat related Samzdat on Kuhn
Ugh, is it in fact true, that I actually agree with apokrisis? Yes this sounds right in the context of this topic.
Quoting apokrisis
I agree with you here up until you said "existence". Your idea about constraints and symmetry-breaking works great in the cases of math and physics. This idea is perfectly ammenable to these topics. This is where your talk of constraints and symmetry-breaking shine and hold the most weight. However, you seem to make the illegal move to apply it to any and every subject in a totalizing fashion.. Besides killing any other angles of inquiry (which would be taking advantage of the open-endedness of philosophy I was talking about) you are quick to dismiss all else to constrain your framework, thus limiting possibilities of other frameworks. But more important than this, you apply such methods/language-games to problems such as the Mind-Body problem. This is where your theory is in deep water and breaks down. Where math is all modeling, you try to overmine the modeling language-game (constraints/symmetry breaking, etc.) to experience itself, and then when people accuse you of never penetrating beyond the models- you defensively go back to the Romantic vs. Enlightenment rhetoric to hand-wave the rebuttal. Your argument becomes a circularity back unto the modeling.
Now, I agree with you very much about your ideas as they relate to math. I have no problem with that move. Its the totalizing of its application to all areas that this becomes questionable.
But the context of that is my own earlier posts in this thread.
Quoting apokrisis
So I was specific that existence = persistence in the face of instability or chaos. I am talking about Peircean process metaphysics.
Quoting schopenhauer1
Illegal? It could be warranted or unwarranted - the evidence can decide the case. But it is illegal to hypothesise?
Quoting schopenhauer1
Hey, do you see me trying to rule out hypotheticals by resort to rhetoric like SX? I welcome your hypotheticals. I just introduce them to the facts of reality. Nature has already chosen what is true. :)
Quoting schopenhauer1
Blah, blah, blah.
You are back to front. Peircean metaphysics begins in phenomenology. And it is not surprising that I arrived at Peircean metaphysics via a dissatisfaction with the prevalent reductionism and epiphenomenalism in mind science (and philosophy of mind).
So Peirce (like Rosen and others) made the deep structural connection that can connect epistemology and ontology.
Mind is a modelling relation. Epistemic fact.
Our model of mind is then going to be a model of this as a suitably general ontic fact. Modelling - or semiosis - is how minds arise in a natural fashion.
Then completing this philosophical trajectory, even matter may be explainable as a pansemiotic fact. Matter exists as an (attentuated or effete) form of the same essential modelling relation ... in some intellectually useful sense.
And guess what. As I keep saying. Physics has gone that way. Everything that exists is the product of informational or holographic constraints on entropic degrees of freedom.
Keep up with science and it is pan-semiotic.
Quoting schopenhauer1
Questions are fine. This is the bleeding edge of metaphysical speculation. You ought to be questioning.
I'm just reminding that I've already replied many times on the same questions. And the criticisms are not penetrating.
I would also note that the reason why Peirce (and all the others I would cite) are getting it right is because they are structuralists, they are thinking in terms of fundamental mathematical basics.
Vagueness, dichotomies, hierarchies - these are all mathematical-strength concepts. They capture the architecture (the architectonics!) of Nature because they begin from first logical principles. They are what symmetry/symmetry-breaking looks like when described in general mathematics.
Now - because they are logical/holistic arguments - they are not the kind of maths you do a lot of calculation with. They are the meta-models rather than domain specific models. But there are then plenty of those kinds of models too now - all the stuff arising out of condensed matter physics, non-linear dynamics, whatever.
So maths itself is doing a better job of describing the structure of nature as it actually is.
As I said, it started out with geometry - existence in space, with time and energy left out of the equation. That was what made the Platonism objectionable - what is real about the form of a triangle?
But replace the bloodless triangle with some real life dynamical flow - like a fractally branching river - and suddenly you really are starting to talk about Nature in a way that has fundamental unifying scope. Suddenly you can see why Nature has to express fractal order so as to be able to exist - or rather, persist as a now regulated and equilibrated source of instability.
I never said a point is the edge of a line. Your putting words in my mouth. An edge marks the boundary of a region, a point marks the boundary of a line segment. A region is two or three dimensional, a line is one dimensional. Why are you intent on producing ambiguity?
Quoting apokrisis
Take away the line and there is just the point. Remember, this is conceptual, we're not talking about a line drawn on a paper or any such thing, we are discussing concepts.
Quoting apokrisis
No, that's absolutely false, a line is not composed of points. Who taught you geometry? A point is zero dimensional, and a line is one dimensional. An infinite number of zero dimensional points could not produce a one dimensional line. This is the incompatibility between the dimensions which I referred to.
Quoting apokrisis
No again, a point doesn't have a location. It is conceptual and concepts do not have spatial location. A point represents a location, it does not have a location. When I say that there is a point which is half way between where I am and where you are, I use "point" to represent this place. It does not mean that there is literally a point existing at this location. When someone says assume a point halfway between A and B on line AB, it does not mean that there is a point existing at this location, making up part of the assumed line. "Point" is used to represent this location.
Quoting apokrisis
Again, that's not true. Geez, what are they teaching in school these days, that kids like you get so mixed up?
Two lines may cross at any random angle, and represent two distinct dimensions. "Orthogonality" is the product of choosing "the right angle" as the distinction between two dimensions. The choice of the right angle (because it had already been proven to be very practical for producing parallel lines), means that any line at a different angle from the arbitrarily chosen two perpendicular lines, is necessarily a two dimensional line. Why not choose that two dimensional line as the representation of one dimension instead? Why are those other, arbitrarily chosen perpendicular lines the privileged signifiers of the two dimensions?
Theoretically, we could assume an infinite number of rays around a point, and assign to each ray a dimension, such that there would be an infinite number of dimensions. That classical "dimensions" are produced by right angles, and are therefore orthogonal is completely arbitrary.
Quoting apokrisis
Utter nonsense apokrisis. Pi is the relation of the circumference of a circle to its diameter, as a measurement, length. That a straight line which marks the diameter is designated as 180 degrees, is irrelevant to the value of pi.
For pity's sake. Can't you see you are just saying what I said?
A line is a 1D edge to a 2D plane. A point is a 0D bound to a 1D line. So you are simply choosing to pretend to be confused by the fact that we use terms that speak to the specifics of some act of constraint.
Yes, a line is an edge to a plane. And a point is only an "edge" to a line. But if you can't see that in the context of my account that the similarity of the nature of the constraint, the form of the symmetry breaking, is exactly the same, then I've no idea how to talk about interesting ideas with you.
Quoting Metaphysician Undercover
You're taking the piss now? Or maybe you are 90+. Seems possible.
Quoting Metaphysician Undercover
And those two distinct dimensions would be distinct because ....?
[Clue: it rhymes with "morthogonal".]
Quoting Metaphysician Undercover
I don't know what they taught you at high school Granddad but you are just imagining any number of rays in a spherical co-ordinate space - a description that is dual or dichotomous to the usual Cartesian one. https://en.wikipedia.org/wiki/Spherical_coordinate_system
If you did go to big school any time in the last century or two, you would have learnt that higher dimensional geometry doesn't work like that. You could indeed have an infinity of spatial dimensions, but they would all have to be orthogonal to each other as that is the critical thing making them a distinct dimension of the one connected space.
Why do 3D knots come undone in 4D? Trying working that one out.
I like this, if I follow you (& I'm not 100% sure I do.) I especially appreciate the example, because I've been trying to quit smoking for a while now. This gives me some solid ground to work from.
[was gonna dig into your post by way of my own attempt to quit smoking] but actually now that I've read through that posts a few times
Clarifying question first:
[quote=fdrake] the usual way people occupy frames constrains variation in their own frame changes by a delimitation of how the other frames are embedded perspectivally into each other. Most don't matter, some matter a lot, sometimes we're surprised by something that didn't matter becoming something (or already was something) that matters a lot.[/quote]
I was puzzling over the first sentence for a while. Just clicked now though, as of writing the last sentence. Is "frame changes" "frame-changes"?
If so, is the gist of the first sentence something like: each of the frames a person occupies tends to somehow refer to - or take into account - the other frames they occupy? So even a frame you're not currently 'in' is, in some way, supported by the one you are in?
I have in mind the image of a group of good friends. The experience ( frame) of hanging out will be different depending on which particular set of friends are present. But no matter what particular set, and no matter how much the experiences of the different sets vary - they still all kind of refer to each other, to the group of friends (frames) itself. You might experience radically different things and think about things in different ways hanging out with a certain person, or group of people, from within that group. But there's always (provided everyone's on good terms) a kind of unspoken awareness of how this experience can connect back to other experiences with the larger group, or different subsets from within that group. (though of course there's griping and shitting-on too.) This is why, when friends hang out, there's some fun in swapping stories about different escapades, which involve different groupings of friends.
So you have a quasi-encompassing frame knit from smaller frames (though that phrasing is off too.) I think you're saying something similar about a single personality, and the variations that personality goes through?
Does that make any sense? The friend thing was really really abstract, I could probably flesh it out with illustrations
But if you follow the example, it's clear that invention and discovery are not so clearly separated; the paper referenced in the OP speaks of (mathematical) creativity as "fall[ing] somewhere between 'invention' and 'discovery'", but I think it's possible to be more precise: we invent because we discover, and we discover because we invent; there's a reciprocal dialectic here; again, follow the example: we 'discover' the irrational, but we're not sure, at first what to 'do' with it. All we know is that it's causing us 'problems': it is a problem (for our understanding of things). And by 'doing' - let me be crystal clear - I'm talking about what kind of sense we want to impart to it (the irrational), how we want to classify, categorise, and think about it: It is a number, or not?
We make a choice. And in so doing, we invent, we create a new, modified concept of number, a concept that might have been otherwise (B&C: "We – users of mathematics, members in a wide sense of the mathematical community – take certain aspects of mathematics to be thus-and-so rather than otherwise"). And now the tricky bit to understand: this inventiveness exerts retroactive effects on the very status of 'discovery': we can only say we have discovered an irrational number to the degree that we have invented a new concept of number that allows the class of irrational numbers to be designated as numbers to begin with (cf. Sauvagnargues: "a contingent irruption (chance) unleashes its own logic, its virtual problem, from which the supposed linearity of prior history is retrospectively configured").
Discovery and invention are co-implicated with each other, each conditioning the other according to a temporal circuit in which discovery prompts invention which in turn conditions the very status of discovery. So the question is not 'is math invented or discovered?', but 'what is the status of invention and discovery when it comes to mathematical concept determiniation?' (or any concept determination whatsoever, I want to argue). What this account is so far missing - what it is necessarily missing - are the pragmatic conditions which 'sway' the choices 'we' (the community of math users) make in one way or another. And these cannot be 'given'; there is no theorem that dictates - within the math - how math ought to be used. So the question which then needs to be addressed is what accounts for the/your intuition - and so far it is only an intution made without proper argument - that "if the game was run again things would work out roughly the same".
First, I think this intuition is probably correct, but perhaps for different reasons to you. 'My' reason would be that the concerns of humans - the things that matter to us, the the things we find significant in life - are probably rather uniform, and would themselves be roughly the same if you 'ran the game again': I can do things if I can figure out the hypotenuse of a right triangle and make it amenable to calculation - perhaps build a house a bit better, construct a rocketship with that much more precision. What would be 'invariant across histories' is not some deep, transcendental structure written in the stars as if by divine diktat - no matter what theologians and pretend-naturalists/fake pragmatists like Apo tell you - but the concerns of living beings with fine-span metabolisms and the need to keep warm: concerns which condition necessity.
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.
Notwithstanding the above, I'll have a stab at explaining the problem-solving algorithm using the example you've given and highlight how it might be different if I was working on a philosophical problem within a prescribed framework, and different again if I was working on a philosophical problem somewhere like a public discussion.
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.
The second step would be to determine what would constitute a solution, what marks the solution I might come up with as being a good one. Again, with framework-prescribed philosophy, the solution would have constraints set by the framework, with public philosophy (I'm thinking mostly of ethics committees here, but I think it applies to this forum too) a 'solution' is a completely different thing, it's more about a perfect mixture of inter-translatable direction, and explanation mixed into one. With maths/science it very different again, the definition of a solution is all about repeatability, can others using your language do what you did and get the same result.
I mean, this is an oversimplification of course, but it might go some way to explaining why I'm not on board with this "maths is like philosophy" paradigm. I think it's like saying cars are like washing machines in that they're both designed to do a task. True, but vacuously true, it doesn't tell us anything about how to design better cars, or how to critique washing machines, for that we need to know what those objects are used for.
Good, you recognize your mistake then, when you said that a point is an edge to a line. That's a start.
Quoting apokrisis
What? Why contradict yourself? That's the end of that start.
Quoting apokrisis
I think that's why your account is unintelligible to me. Things which are different, you claim are the same. You'll insist that it's a difference which doesn't make a difference, but that's nothing more than contradiction. And your whole account of symmetry breaking is based in contradiction.
Quoting apokrisis
The dimensions are distinct because they are designated as such. That designation is based in the assumption that there is an angle which distinguishes them one from another. The assumption of an angle is completely fictitious. So any distinction which separates one dimension from another by employing a specified angle is a fictitious distinction and is therefore completely arbitrary.
Quoting apokrisis
Right, and the point being that there is an infinite number of possible ways to construct "dimensions" which lie between the orthogonal way that you imagine it, and the way that I just imagined it. My argument is that each of them will end up with the very same problem of incompatibility (irrational numbers) between one dimension and another, because each utilizes the same falsity, the angle. Until we get rid of this antiquated way of modeling spatial existence, with dimensions and angles, we have no hope of producing a proper understanding.
Quoting apokrisis
That spatial dimensions must be orthogonal is just a convention. You break the conventions, saying that a point is an edge, so why can't I break the conventions? Furthermore, your breaks in convention result in contradiction, because you're careless, mine do not result in contradiction, because I'm careful to analyze what I am doing.
If I may, the entirety of Wittgenstein's Philosophical Investigations can be read as nothing but a critique of incomplete problemscapes; its alternative title might have well been: A Critique of Pure Problems. Every single word in that book can be considered a critique of badly-posed questions, and as a hand-guide as to how to pose questions well.
§380: "How do I recognize that this is red? a “I see that it is this; and then I know that that is what this is called.” This? a What?! What kind of answer to this question makes sense? (You keep on steering towards an inner ostensive explanation.) I could not apply any rules to a private transition from what is seen to words. Here the rules really would hang in the air; for the institution of their application is lacking".
And then there are the Lectures on the Foundations of Mathematics: "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."
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.
Quoting 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 [math]\small \exp(-x^2 /2)[/math]... 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.
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.
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.
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.
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 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 [math]\exp(x^2 /2)[/math]). 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.
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.
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.
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.
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'.
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.
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?
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.
Quoting fdrake
How have you concluded that it's supposed to be an impossibility? I can't see how this is implied.
Quoting 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.
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.
By definition, that fulfills the criteria of solipsism. Don't you agree?
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?
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,
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.
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).
Really? Then I must apologise for hijacking such an entirely mundane discussion with a side-track into the authority by which we judge philosophical propositions. That we can abstract enough to discuss the presumptions essential to our discourse (or not depending on your view of hinge propositions), seems, as my comment implies, self-evidently true. I'm not sure what else a thread entirely about it might be trying to say.
The statement I was arguing against was this one in the OP;
The paper makes the point that maths is governed by choices and proofs sometimes simply 'fall out' of making those choices, other times, as with SLX's example above, the proofs are simply chosen or ignored out of preference for what they can do.
But the argument that philosophy is "...just like this", has been advanced without further explanation as to why.
The first part, that philosophy too makes choices about frames which in turn dictate solutions, is pretty uncontroversial. The second part, that is does so out of some quest for utility, their "their fruitfulness for ... whatever it is we want to do" remains an entirely unsupported assertion. What exactly is it that philosophy "does" in this argument, that frames could be chosen on the basis of their ability to further? In what way can any philosophy be rejected as 'bad' philosophy, under this understanding? Presumably, because it isn't fruitful at producing whatever it is it's supposed to produce. And yet we have no evidence to show that this is happening at all. Hence my comment 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."
This is the whole point of Wittgenstein's investigation. A point not lost on the authors of the paper themselves who note the importance of Wittgenstein's solutions to the rule-following paradox.
It's rarely the actual preliminary points made in any of these threads that I object to. Largely they are trivially true. It's the game they're used to play, it's the "therefore...." that comes at the end of it all that bothers me.
Language on holiday. Exemplary of the kind of pseudo-question that philosophy rightly rejects.
But it's easy to answer your question, I don't understand the difficulty. A machine does a job of work. If you're trying make a connection with utility, then you need a person. Utility requires a person to be useful to, nothing can be useful objectively. A machine is useful if it does work that the user of the machine wants it to do. There's no requirement that I judge its utility in order to define it as a machine, but there is in order that I judge it as a 'good' or 'bad' machine.
I don't see what point you're trying to make here.
It's a machine if it does some job of work, it's a useful one if it does that job to the satisfaction of the person using it.
It's philosophy if it makes some argument about 'the way things are' that cannot be checked against objective empirical sense data (or whatever definition you prefer). It's a useful one if the person holding it finds it satisfying.
The problem comes when someone who does not find an argument satisfying tries to claim that it is therefore universally 'bad' by some hashed-together criteria because they can't handle the relativism of philosophical propositions.
Wittgenstein did not provide an adequate description of what it means to follow a rule. He stated that if one could be observed to be acting in a particularly described way, then that person could be said to be following a rule. But this requires that the person do the same type of thing, more than once, producing an inductive conclusion by the observer, in order that the person is following a rule. Therefore "following a rule" is a property of the inductive conclusion of the observer, rather than a property of the person following the rule.
In reality, to follow a rule is to hold a principle within one's own mind, and adhere to it. This allows that one is following a rule in the very first instance of acting according to the rule. Furthermore, it allows for the very important, and relevant type of following a rule, which is to restrain oneself from a certain activity, like we do with a resolution to quit a bad habit. This is clear evidence that Wittgenstein's description of rule following is way off the mark. If one is successful in quitting a bad habit, there is no second time, and according to Wittgenstein this person could not be following a rule..
So Wittgenstein doesn't really solve any "rule-following paradox". All he does is define "rule-following" in a way which could make the problem appear to go away. But his definition doesn't match what rule-following really is, so what he has done is produced a fantasy solution which is really quite useless because what it resolves, if anything, is an issue with what Wittgenstein refers to with "rule-following". and this is nothing real.
"It's philosophy if I'm philosophically incompetent".
Interesting example of what I'm talking about.
Quoting Metaphysician Undercover
But he obviously did. I don't think there can be any doubt that Wittgenstein was a very clever man. He obviously found it adequate, as do a number of equally clever Wittgenstein scholars who still hold to his solution to a greater or lesser extent. (one could include John McDowell, Simon Blackburn, Saul Kripke, potentially Crispin Wright). So unless you are privy to some unique insight these other scholars lack, one of two things must be the case - either one group is wrong but it will be impossible to tell which (all the relevant data having already been presented), or you are simply using words differently to describe the same thing. On no account does the mere presentation of a counter-argument demonstrate anything at all about the 'adequacy' of Wittgenstein's solution other than an expression of your own personal satisfaction with it.
Its possible to arrive at dozens of counter-arguments to your position, not that doing so makes your position wrong either. We could say that the first instance is not a true expression of 'rule-following' (having just invented the term, we're free to define it as we see fit), we could justify such a distinction by saying that the first instance represented an investigation, whereas only subsequent ones can be said to follow a truly 'private' rule. We could claim that one could not be said to follow a private rule until they had personal experience which removes it from the public sphere. And on and on. At no point in time is anyone 'proving' to anything. Nothing is what is happening "in reality" because we do not have unfiltered access to 'reality'.
If, for some reason, you feel some need to be abundantly clear about your world-view with regards to the solutions to the rule-following paradox, then I'm sure that rendering them available for criticism is a good way to refine them to your satisfaction, but personally it's a point I'm happy to accept a number of possible interpretations of.
This seems to miss the point of Wittgenstein's challenge regarding rule following (at least under Kripke's interpretation of it) - it merely pushes the sceptical challenge back to asking what tells you which principle it is that you hold in your own mind.
Sure, Wittgenstein found his description of "rule-following" adequate for his purposes, and perhaps these other philosophers found it adequate for their purposes as well. However, as I explained, it doesn't apply to a vast quantity of instances of rule following, therefore we would be foolish to accept it. Would you accept a description of "plant" which was inapplicable to a large number of things which we call by that word? I would reject the definition as unacceptable, wouldn't you?
In any situation where an observer was incapable of identifying the rule which an individual was following, as a specific rule, the individual could not be following a rule. That means that a large number of cases where a person is actually following a rule, we have to say that the person is not following a rule. Furthermore, if a person makes a rule, and follows this rule with one's actions, but the actions are inconsistent with, or contrary to a rule identified by an observer, the observer would have to say that the person is not following a rule, because the person is breaking a rule. However, that person is clearly following a rule, it's just that the observer has wrongly identified the applicable rule, saying that the person is breaking a rule, rather than that the person is actually following a different, unidentified rule.
Quoting Pseudonym
It's very easy to see who is right and who is wrong. My description applies to all cases of "following a rule" whereas Wittgenstein's only applies to some specific cases. According to Wittgenstein's description, a person is only following a rule if the person acts in the right way. This excludes the possibility that a person who is acting in the wrong way is actually following a rule. So all the instances when a person is acting in the wrong way, yet is still following a rule, are excluded as instances of rule following. It is obvious therefore, that Wittgenstein's rule, concerning rule following, is wrong, because it disallows the possibility that when someone is wrong, they are still following a rule. It only allows us to say that a person is following a rule if the person is judged to act in the right way, despite the evidence that people who act in the wrong way are still following rules.
Quoting Pseudonym
Right, I was very dissatisfied with Wittgenstein's proposed solution. And if you follow what I say, then unless you have a rebuttal for me, you ought to be dissatisfied with it as well, regardless of whether any other philosophers are satisfied with it.
Quoting Pseudonym
Sure, you can define "rule-following" however you please, and offer this to me. It doesn't serve as a rebuttal though, because my argument is against the use of such a definition. What is the point in defining "rule-following" such that it excludes a vast number of instances which we refer to as following a rule? If it's just for the purpose of solving some paradox, then the paradox is not really solved, because all these paradoxical instances of rule-following are still going on, despite the fact that your definition of "rule-following" denies that we can call these instances of rule-following.
Quoting jkg20
I really don't see your point. Care to explain?
This is presumptuous. There are developments in quantum mechanics - extending to its ontology and thus the formal logic used - where quantum objects lack identity. They are not self-identical, sometimes terms "non-individual objects". Also, there are non-classical logics (and thus corresponding non-classical mathematics) where contradictions can be proved without trivialism, thus allowing one to prove the existence of inconsistent mathematical objects (e.g. the Russell Set).
No, you didn't "explain" it, you asserted it using your private definitions;
"In reality, to follow a rule is to hold a principle within one's own mind, and adhere to it." - Which 'reality' are you referring to here, and what god-like insight has allowed you to simply 'know' what it is to follow a rule in it? Philosophers can't even agree what a mind is, let alone what's in it. Peter Hacker, for example, doesn't even think there is such a thing as a mind.
"This allows that one is following a rule in the very first instance of acting according to the rule." - Why would we want, or need, to allow this?
"...it allows for the very important, and relevant type of following a rule, which is to restrain oneself from a certain activity, like we do with a resolution to quit a bad habit." - Again, why would we want, or need, to allow this?
"... If one is successful in quitting a bad habit, there is no second time," - Tell that to an alcoholic, most consider their entire lives to be a continual struggle to not drink alcohol.
"That means that a large number of cases where a person is actually following a rule, we have to say that the person is not following a rule." - How do you know they are "actually" following a rule? The whole point of the paradox is that we have no way of defining what it is to "actually" follow a rule.
Quoting Metaphysician Undercover
Yes, of course I would have to. If the rest of the speaking world were referring to some object as a 'plant' which I personally considered not to be one, or vice versa, I would have to follow suit in order to communicate. The is no thing that 'plant' means outside of its use. You're arguing that your personal uses of the the term 'rule' need to be included in the global definition of what it is to follow a rule. That's the whole of what Wittgenstein had to say about Private Language.
Quoting Metaphysician Undercover
This is simply wrong, in that this is not what Wittgenstein said. His claim was that we would have no way of knowing whether a person was following a rule correctly causing their actions or following a different rule but making a mistake.
Quoting Metaphysician Undercover
It's not about following a rule it's about the inability to know which rule a person is following.
But this is not the right place to get into a deep discussion about Wittgenstein's rule-following paradox. It is relevant to this thread, as the authors of the paper in the OP point out, in that one cannot say anything concrete about solutions arising from framework choices because one cannot say anything concrete about what rules the respective thinkers are actually following to derive their conclusions.
I follow rules all the time, don't you? I hold a principle within my mind and adhere to it. There is no "god-like insight" involved in me knowing this, just a little bit of self-reflection. It's really quite straight forward, you ought to try it sometime. However, you for some reason seem to think that following a rule is some sort of complex, and difficult thing to understand, requiring a god-like insight. Why make it so difficult when it's not?
Quoting Pseudonym
Right, so my argument is that Wittgenstein didn't account for a vast amount of usage of "rule-following" when he defined it. So he acted in a hypocritical way, arguing that usage must be accounted for in producing a definition, but then not doing that when he produced a definition for rule-following.
Quoting Pseudonym
This is a misunderstanding of what I said. I was not talking about a situation of when a person appears to be following a rule, but is really not following that rule, I was talking about a situation when a person appears not to be following a rule, but really is. These are the situations which serve as evidence that Wittgenstein's description of rule-following is unacceptable. These are the situations in which rules come into existence. A person thinks up a rule and starts following it. In these situations there is also "no way of knowing" that the person is following a rule, but it must be concluded according to the definition, that the person is not following a rule. This is an unjustified conclusion.
This unjustified conclusion has extensive epistemic consequences. It leaves us with no principles whereby we might judge rules themselves, as right or wrong. A person is judged to be acting rightly or wrongly, according to whether one is acting by the rule, but a rule is not the type of thing which can be judged as right or wrong. This, what you call "no way of knowing" the actual rule which is being followed by the individual, makes it impossible to judge the rule itself. All we can do is judge the individual's actions in relation to known rules. If the person does act according to known rules the person cannot be acting rightly.
Of course there is a simple solution to this problem which Wittgenstein creates, and that is to recognize that judgement, as well as rules, occur within peoples' minds. This allows that an individual, with one's own mind may judge a rule as right or wrong, as well as judging another person's actions as right or wrong. But Wittgenstein's principles leave us without the capacity to judge a rule as right or wrong.
That's the point of this thread, without that capacity, the capacity to judge a rule, rules must be completely arbitrary. However, we actually often judge rules. The op adds "aim", purpose, such that rules are judged and shaped toward specific goals. Now we have to account for the existence of such goals, and this brings us right back into the minds of individuals. So we are no further ahead. Instead of having principles for judging a rule as right or wrong, which is what Wittgenstein avoided the need for, we now need principles for judging a goal as good or bad. Wittgenstein provides us with nothing but a deferral of the problem, veiling the deferral as a proposed solution.
Quoting Pseudonym
I agree, but this "inability to know" is exactly where Wittgenstein makes his mistake, what I called the unjustified conclusion. By Wittgenstein's principles, if there is an inability to know whether or not the person is following a rule, we must proceed as if the person is not following a rule. That is simply how Wittgenstein defines "rule-following", and we must adhere to the definition in our procedure. But this is to proceed on an unjustified premise, and that's why the definition is unacceptable.
Quoting Pseudonym
I don't think you are adhering to Wittgenstein's principles here. By his principles, if we cannot say anything concrete about the rule which a thinker is following, we must conclude that the thinker is not following a rule. This is the critical point which renders Wittgenstein's principles ineffectual for dealing with instances of creative thought. Such choices are left by Wittgenstein as arbitrary, unruly. The op turns to "aim", purpose, to deal with these choices. This puts us back into the minds of the thinkers, which is what Wittgenstein was trying to avoid.
So what conclusion do you think someone with false memory syndrome would come to about what rule motivates their actions? What about phantom limb syndrome, Capgras delusions, synathesia, or simple dementia. How are you so sure your brain serves you up an accurate report of what is it to follow a rule, not just for you, but apparently for all humanity?
Are you implying that Wittgenstein and all the well-respected experienced thinkers who follow his line of argument have all failed to do even a "little bit" of self-reflection? I mean, I don't even completely go along with Wittgenstein (or Kripkenstein) on this issue, im just trying to point out how unlikely it is that such intelligent people are categorically 'wrong' about an issue in respect of which they are in possession of all the relevant facts.
Quoting Metaphysician Undercover
How many people have you spoken to about what it feels like is going on when they use the term "rule-following"? I mean, out of the 7 billion people currently speaking to each other about their experiences, how many of them have you interviewed to arrive at this "vast number" who are using the term and meaning by it exactly what you describe.
Quoting Metaphysician Undercover
It needn't make any difference to the argument. What, to you, is "not following" a rule might well be to Wittgenstein "following a rule which mandates the opposite of the rule you claim the person is" not following". It all just comes down to how you use your terms.
Quoting Metaphysician Undercover
Do they?
Quoting Metaphysician Undercover
No, as I said, it's not about concluding that they're not following a rule, it's that we must conclude that we cannot know what that rule is.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Must we really engineer our epistemology just in order to preserve our ability to judge things as right and wrong. Are you really so beholden to your Minoan complex that you disregard every proposition that doesn't allow you to judge things 'right' or 'wrong'?
Quoting Metaphysician Undercover
I don't see how you've arrived at this conclusion. We need not conclude that the thinker is not following a rile. The Skeptical solution is that we can't know what the rule he's following is, so we can't draw any conclusions about things like consensus, or public meaning. It doesn't imply anything about what we "must" conclude. Also, you've suddenly introduced the idea that instances of creative thought are something we must "deal with". Deal with in what sense?
Quoting Metaphysician Undercover
Exactly. Despite the evident pleasure people seem to get out of it, the idea that some particular notion could be objectively "not useful", "not necessary" or any other term dug out of the thesaurus to avoid saying "wrong" just does not seem to me to be justifiable.
I don't see how this is relevant.
Quoting Pseudonym
Oh, so you don't "completely go along with" it. Then why not give it up as unacceptable? Why gloss over the unacceptability? Who cares if other intelligible thinkers accept it. If you can't go along with it, then take the aspects which appeal to you and leave the rest as unacceptable.
Quoting Pseudonym
What I describe is instances of "rule-following" which are inconsistent with Wittgenstein's definition. These are situations such as when someone resolves to do something, like a New Year's resolution. It appears to me like there is a vast number of people who talk about New Year's resolutions, don't you agree? Or are you going to obstinately insist that following a New Year's resolution is not an instance of rule-following.
Quoting Pseudonym
This is a misunderstanding of Wittgenstein. It is very clear that if we cannot know the rule which one is following, we cannot say that the person is following a rule. If the rule cannot be identified then the person must be said to be not following a rule. This is the crux of the private language argument. An unidentifiable rule is not a rule at all. If it were a rule then there could be private rules and private language. So if we conclude that we cannot know what the rule being followed is, then it follows that there is no rule being followed. An unidentifiable rule is not a rule. A private rule would be a rule which cannot be identified and therefore it is not a rule at all. So there is no such thing as a private rule.
Quoting Pseudonym
Reconsider your misunderstanding of Wittgenstein which I described above. A private rule is not a rule. If the thinker is following an unidentifiable rule, the thinker is not following a rule. This is fundamental. Have you read the Philosophical Investigations, and how Wittgenstein describes what it means to follow a rule? Or, are you going by some secondary source, a watered down version with an author trying to cover up the unacceptability of Wittgenstein's proposal? Why would someone try to make an unacceptable principle appear acceptable?
Really if all you've got to say is that your interpretation of Wittgenstein is right and mine is wrong without any attempt to substantiate that claim then there's little point in continuing this discussion. It consistently astounds me how many posters on this forum seem to have not the faintest clue as to how diverse, contradictory and most times mutually exclusive, propositions and interpretations are in academic philosophy, particularity about Wittgenstein. I suppose it takes a certain hubris to think that one can contribute meaningfully to a discussion and so that may filter out those who see the diversity of opinion as a probably the only true fact out there, but still, the level continues to surprise me.
Quine and Kripke have different interpretations of Wittgenstein's intent in removing facts about meaning. Scott Soames at Princeton disagrees with both. Cripin Wright sees a fairly full removal of judgement about rules, John Mcdowell disagrees, seeing no problem with consensus judging. If you can think of a possible way to interpret Wittgenstein, chances are someone's written it. I have not, however, yet read any interpretation of Wittgenstein that suggests that he is making the claim that we must conclude the thinker is not following a rule at all.
John McDowell has the most sympathetic interpretation to your preference that we maintain some ability to judge right and wrong, yet he specifies "Wittgenstein's target is not the very idea that a present state of understanding embodies commitments with respect to the future[rule-following itself], but rather a certain seductive misconception of that idea."
Even the more Skeptical interpretations of Crispin Wright only go as far as to say "Wittgenstein seems almost to want to deny all substance to the 'pattern' idea" (not even my emphasis, he put that in).
Or, from Investigations itself (219) "“All the steps are really already taken” means: I no longer have any choice. The rule, once stamped with a particular meaning, traces the lines along which it is to be followed through the whole of space. – But if something of this sort really were the case, how would it help? No; my description only made sense if it was to be understood symbolically. – I should have said: This is how it strikes me. When I obey a rule, I do not choose. I obey the rule blindly."
If there is an interpretation where Wittgenstein insists we must presume the thinker is not following a rule at all, however, it would not surprise me in the least. Why don't you actually quote the passage you think is making that claim (or the secondary interpretation) and we can look at it.
Or you could just join in the general condescending approach to philosophy here that every position you don't personally agree with must be the result of a deep ignorance of the subject on the part of your opponent. I'd just rather not continue under this second approach.
Presuming the former approach, pending your posting the actual quotes you're using for your interpretation, I will quote a few sources for mine;
Consider first this passage in the PI in which Wittgenstein reassures his interlocutor:
"But I don't mean that what I do now (in grasping a sense) determines the future use causally and as a matter of experience, but that in a queer way, the use itself is in some sense present." - But of course it is, 'in some sense'! Really the only thing wrong with what you say is the expression "in a queer way". The rest is all right ...."
Wittgenstein is not saying that a person does not follow a rule (this seems obvious to me from the passage at 219 where he clearly describes his feeling about the process), but that there are no facts about that rule which can be used to determine if it is being 'correctly' followed. As John McDowell puts it in 'Wittgenstein on Following a Rule', "these natural ideas lack the substance we are inclined to credit them with", or Crispin Wright saying a similar thing "there is in our understanding of a concept no rigid, advance determination of what is to count as its correct application" in his paper 'Rule Following without Reasons' (if you haven't read it by the way, it is an excellent summary of an interpretation I have a lot of sympathy with, but I'm not sure if it's available on the internet)
It is his understanding of what Wittgenstein undermines which was the reason I mentioned the whole rule-following thing in the first place (which I think may now be off topic?) He says that the rule-following problem undermines the notion that "Discoveries in mathematics are regarded as the unpacking of (in the best case) deep but (always) predeterminate implications of the architecture of our understanding of basic mathematical concepts, as codified in intuitively apprehended axioms.", @StreetlightX's position (that this recognition does not remove our ability to judge) simply tries to shift the rule-following to another dimension - how things should fall out of frames, how frames are or are not useful etc. That is the the only sense in which this interpretation of Wittgenstein on rule-following is relevant to this thread. Crispin Wright again clarifies perhaps better than I could "no response, however aberrant, in and of itself defeats the claim that a subject correctly understands and intends to follow a particular rule – you can always make compensatory adjustments by ascribing a misapprehension of the initial conditions for the application of a rule, as expressed in the minor premise in the modus ponens model" (which he explains earlier likening rule-following to a simple chess game).
If you want to discuss rule-following in general then perhaps a fresh thread might be appropriate? If you still see a relevance to the interpretation of the paper in the OP, then I'm not quite following and perhaps you could make that link clearer. Alternatively, If you'd like to just like to continue the presumption that every opposing position must stem from naive ignorance, then you would, it seems, at least have some company.
Quoting Pseudonym
My claim is not that we must conclude that any given thinker is not following a rule. It is that if a rule cannot be identified from the person's actions, then we must conclude that the person, despite believing oneself to be following a rule, is not following a rule. I really can't understand why you argue against this, as it is the key premise to the so-called private language argument. Are you arguing that those Wittgensteinians who produce a private language argument from his principles have misinterpreted him? Here's a simplistic version of the private language argument. If you think that you can create a rendition without the third premise, the one that you claim Wittgenstein didn't state, then show me.
P1. Language requires following rules
P2. A private language would consist of following private (unidentified) rules.
P3. Following private (unidentified) rules do not qualify as "following rules".
C. Therefore a private language is not possible.
Quoting Pseudonym
All right, here you go
[quote=Philosophical Investigations]201. This was our paradox: no course of action could be determined
by a rule, because every course of action can be made out to
accord with the rule. The answer was: if everything can be made out
to accord with the rule, then it can also be made out to conflict with it.
And so there would be neither accord nor conflict here.
It can be seen that there is a misunderstanding here from the mere fact
that in the course of our argument we give one interpretation after
another; as if each one contented us at least for a moment, until we
thought of yet another standing behind it. What this shews is that
there is a way of grasping a rule which is not an interpretation, but which
is exhibited in what we call "obeying the rule" and "going against it"
in actual cases.
Hence there is an inclination to say: every action according to the
rule is an interpretation. But we ought to restrict the term "interpretation"
to the substitution of one expression of the rule for another.
202. And hence also 'obeying a rule' is a practice. And to think one
is obeying a rule is not to obey a rule. Hence it is not possible to obey
a rule 'privately': otherwise thinking one was obeying a rule would be
the same thing as obeying it. [/quote]
Notice 202. It explicitly states "And to think one is obeying a rule is not to obey a rule." This is what makes it impossible to obey a rule privately. Holding a principle within my mind, and adhering to it is to "think I am obeying a rule". But this is explicitly "not to obey a rule". That is what Wittgenstein excludes from "obeying a rule". As a result, "obeying a rule" is restricted to a practise which is observed to be in accordance with a rule. If the observer cannot identify the rule, because it is only the actor who holds the rule within one's mind, this is "to think one is obeying a rule", which is "not to obey a rule".
Which is why your conclusions about how we must respond to the rule-following paradox are not necessitated by it. As I said, There are numerous interpretations, there's no 'right' or 'wrong', there's no 'unacceptable' it just depends what conclusion you want to come to and then re-arrange the meanings of the terms to suit. The whole philosophical argument resulting from the rule-following paradox is about how we conceive of 'a rule', not about how we 'must' respond to others in respect of whether they are following one or not, that is part of the paradox, not one of the the solutions to it. If the private rule-following behaviour is or is not really 'rule-following', then does that mean anything? It's certainly some form of behaviour. It's undeniably a different form of behaviour to following public rules, so what difference does it make if we call it 'rule-following' or not?
In Remarks on the Foundations of Mathematics, Wittgenstein says "How was it possible for the rule to have been given an interpretation during instruction, an interpretation which reaches as far as any arbitrary step?” (RFM VI-38), which I think explains his position more clearly than it is expressed in PI. Basically, he's saying that if a rule must be interpreted 'correctly' to be followed 'correctly', then where is the 'correct' interpretation? It can't be in the rule itself (that would be self-referential), so it is nowhere.
Either there is no 'correct' interpretation because we have no reason to believe rules are other than as they appear to us - the quietism of McDowell, or it is not the case that rules must be 'interpreted correctly' in order to be followed correctly, or there is a 'correct', but we cannot know it (the Skeptical solution)
So, you'd said...
Quoting Metaphysician Undercover
But this is not what 'must' be concluded at all. This is part of the paradox. It's conclusions are the rejection of private language and either quietism, skepticism or (if you must!) the anti psychologism of McGinn.
I'm not sure if I'm explaining it any more clearly, but I will try to use the example you gave of a person quitting smoking. You'd said that such a person must be following a rule - the rule "I will not have a cigarette", but that it is perverse to say he's not following a rule simply because we cannot say if he is following it correctly by his action of not having a cigarette. So this is the paradox. But Wittgenstein says that we cannot simply say he is following a rule (one in his own mind) because even he does not know all the interpretations of that rule until they arise, he has not, for example specified whether, should some company invent a new type of smoking device, that constitutes 'a cigarette' or not. He has not defined 'cigarette' against all possible future issues, nor could he ever define 'cigarette' without using other words which he would then have to define...and so on. So either we must remain quiet on whether the man is following a rule (or breaking it), or we must conclude that he might be but it's impossible to know. Or, to use Crispin Wright's words instead;
"If a (suitably precise and general) rule is—by the very definition of 'rule', as it were—intrinsically
such as to carry predeterminate verdicts for an open-ended range of occasions, and if grasping a
rule is—by definition—an ability to keep track of those verdicts, step by step, then the prime
question becomes: what makes it possible for there to be such things as rules, so conceived, at
all? I can create a geometrical figure by drawing it. But how do I create something which carries
pre-determinate instructions for an open range of situations which I do not think about in creating
it? What gives it this content, when anything I say or do in explaining it will be open to an
indefinite variety of conflicting interpretations?"
That, at 202 is the premise, the definition of "obeying a rule" which allows for the private language argument. As you can see, it's very clearly stated:
"And to think one is obeying a rule is not to obey a rule. Hence it is not possible to obey a rule 'privately':"
I really don't care what "most scholars believe", I know how to read. I am doubtful about some scholar's ability to do that. As I said, if you think that you can formulate a private language argument without that premise, then demonstrate it. Otherwise you're just blowing smoke.
Quoting Pseudonym
What are you talking about? Either you continue to deny that Wittgenstein stated this principle or you accept that he did, and accept what the proposition means, as well as the logical implications. You cannot accept the proposition, and the private language argument which follows from it, yet reject what the proposition means.
I reject the proposition because I do not accept the meaning of it as representing what I consider to be rule-following. You seem to agree with me, but instead of rejecting Wittgenstein's principles, you insist that Wittgenstein didn't mean what is explicitly stated. Then what do you think Wittgenstein means with the conclusion, that there cannot be a private language?
Quoting Pseudonym
Right, it is Wittgenstein's definition of "rule". To think that I am following a rule is not to follow a rule. As I said, this definition excludes all the times that I hold a principle in my mind, privately, and follow that principle, as a rule. This is very explicitly "not to obey a rule". Either we can accept this definition, or we can reject it as unacceptable. But for you to try and say that Wittgenstein didn't mean this, what is stated so explicitly, is complete nonsense.
Quoting Pseudonym
Nonsense, it is explicitly stated, he is not following a rule. This is the way that the paradox is avoided, and private language is denied. We do not have to say whether the person is following the rule correctly or not, because the person is simply, and very explicitly, not following a rule. And, the consequences of this, are the absurdities I referred to earlier, concerning the relationship between rules, and right and wrong.
It's not difficult. It's only difficult if you desire to hide this premise, because it's unacceptable to you, yet you also want to maintain the conclusion of the argument. That's deception though.
Well that's relatively simple since the private language argument only really starts at 243, the section on rule-following being only one of the numerous preliminaries to it in PI.
The argument traditionally is expounded using definitions of signs.
1. A definition of a sign cannot be formulated privately because to define a sign is not simply to associate it with a sensation at the time, but to correctly associate with that sensation at all times. As Wittgenstein says "“I impress [the connection] on myself” can only mean: this process brings it about that I remember the connection right in the future’. For I do not define anything, even to myself let alone anyone else, by merely attending to something and making a mark, unless this episode has the appropriate consequences."
2. That such an association would have no meaning (make no sense) as a definition because it could not be doubted (being a private sensation).
Neither explanation has anything much to do with the rule-following paradox. Not that your definition of the 'impossiblity' of following a private rule is a necessary interpretation there either. The exposition of the rule following argument is simply that since you can think you're following a rule when you're not, thinking you're following a rule cannot be the same as actually following a rule. Yet privately (in the sense Wittgenstein uses the term), thinking you're following a rule is all you have, so it's impossible to claim you're following a rule privately.
I get the sense from the tone of your responses, however, that you're not particularly interested in the complexities of interpretation, but rather "he said X, by which he meant Y, and anyone who disagrees must be an idiot."
I really have no more stomach for these types exchanges.
I don't see any argument there. Since we can and do communicate without definitions, your talk of definitions is irrelevant. Definitions are not a required part of language. That sort of private language argument fails for that reason.
Quoting Pseudonym
Right this is the point I dispute as unacceptable. I think that following a rule is nothing other than thinking that you are following a rule. It is to hold a principle in one's mind and adhere to it. That's what "following a rule" is. And, I do not see how it is possible that while one is holding the principle in mind, and adhering to it, (thinking you're following a rule), that person is actually not, as Wittgenstein suggests here. Yes, mistakes are possible but this occurs when we do not hold the principle, or do not adhere to it. At this time, it is impossible to be thinking that you are following the rule, because thinking that you are following a rule is to hold the principle and adhere to it.
As I said before, Wittgenstein switches what it actually means to follow a rule, hold the principle in mind, adhere to it, and act accordingly, with what it means to be judged as following a rule, to act as if one is obeying a particular rule. He has no description of what it means to follow a rule, only a description of what it means to be judged as acting as if one is following a rule. So he dismisses a true description of rule-following, to replace it with a description of acting as if one is following a rules. He does not describe the object, (rule-following) but a representation of the object (act like one is following a rule).
Are you talking about consciously thinking about the rule?
When you first learned how to play chess, you had to do that for a while, but no one who's played for a while ever thinks about the rules while they play, do they?
This is the point. We do not have two minds. We cannot simultaneously hold a view on what a rule is and faithfully, with good intent, make a mistake in applying it. If the rule is indeed private (which means not just known only to us, but knowable only to us), then every action we take is a faithful attempt to interpret that rule for the circumstances we're faced with and a rule is nothing other than it's interpretation in certain circumstances. How can you think you are following a rule (which is known only to you) and yet not be (make a mistake)? Where, and in what form, is the rule kept in your mind which is something other than the responses to circumstances you're faced with?
Consider Srap's chess example above, but imagine a private version. A game which you invented the rules for and only you know them. In this game some piece (which only you know), moves in some way (which only you know), but playing it in your mind you make a mistake you move it in a way that is 'wrong'. How do you know you've made a mistake? How do you know that the piece wasn't actually supposed to move that way and you've misremembered the way you originally intended for it? How do you know that whatever sensory or internal input is telling you that the piece is in the 'wrong' place is the same or different to the one you had when you invented what the 'right' place for it should be? Since you cannot define a rule in your mind other than by the actions that should be taken in response to certain circumstances, you are beholden to the inconsistency of your understanding of 'action' and your meaningful interpretation of 'circumstances' neither of which you can have any faith in. And this is just one simple moving rule in a made up game. How much more unreliable will it be when we come to rules about the meanings of words or ethics?
That's how mistake is possible.
Quoting Pseudonym
One problem of mistake is as Srap indicates. The rule is relegated to memory, and we act most times by habit without consulting the rule. Actions of habit must be distinguished from actions of following a rule.
Quoting Pseudonym
As I described, it is impossible to follow a rule, and simultaneously make a mistake. That's contradiction. If one holds a principle and adheres to it, one is not making a mistake. if one makes a mistake, one is not holding a principle and adhering to it. However, we find ourselves in vastly varying situations, in which we need to interpret the situation, as well as interpret the rule in a way which is applicable to the situation. Mistake is often attributable to misinterpretation. As you can see at 201, Wittgenstein attempts to remove the importance of interpretation. Since interpretation is a major source of mistake, this procedure is unacceptable.
Quoting Pseudonym
In that instance, it's easy to know you made a mistake. You go back and revisit the move while holding the rule in your mind, and see that you made the move absent mindedly.
Quoting Pseudonym
These would be cases of misinterpretation. And, as you describe, in these cases you do not know whether or not a mistake was made. That's life. We cannot liberate ourselves from the restrictions imposed by the facts of life, by changing the definition of rule-following, as Wittgenstein tries to do..
Quoting Pseudonym
Uncertainty is an essential aspect of living as a human being. So be it.
Acting out of habit without consulting the rule would not be a case of the kind of mistake Wittgenstein is talking about. He's talking about a case where one has very consciously tried to apply the rule but nonetheless made an error. It is the impossibility of this kind of mistake which leads to the paradox. It is obviously possible to have what we think is a rule in mind and then not follow it (either deliberately, or absent-mindedly), what is not possible is to think that you are following your private rule when in fact you are not. This, Wittgenstein concludes, must mean that there is no 'fact' of the rule other than your thinking of the following of it at any one time.
Quoting Metaphysician Undercover
But what would constitute a mistake. If your rule was, I must not smoke cigarettes, and a new cigarette-like device entered the market, how would you know whether smoking it was breaking your rule or not? You need to interpret the new cigarette-like item, but how could you possibly make a mistake in that interpretation? Remember, Wittgenstein is talking only about private rules, one's which cannot be verified by checking against public definitions.
Quoting Metaphysician Undercover
Again, we're talk about a mistake made in good faith, not absent-mindedly. One made despite your best intention to follow the rule.
Quoting Metaphysician Undercover
Obviously this is going nowhere. In my mind (and that of most interpretations of Wittgenstein) it is exactly because of the restrictions imposed by the facts of life that Wittgenstein reaches the conclusions he does.
This would be an error of misinterpretation then. Either the situation is not interpreted properly, or the rule is not interpreted properly. So the rule is not applied correctly.
Quoting Pseudonym
So it is very possible and common to think that you are following your private rule, when you are not, because you have misinterpreted the situation, misjudged, and therefore wrongly applied your rule. There is no paradox, the supposed paradox is artificial, made up by Wittgenstein to support his intent to avoid the matter of interpretation. All such mistakes, those which are not due to memory, absent mindedness, habit, can be understood as matters of interpretation. I would divide the source of error into two distinct sorts, errors of not having the rule properly in mind when acting, and errors of interpretation. All possible errors fall into these two categories and there is no paradox to be resolved.
Quoting Pseudonym
I don't see the problem. You must interpret the thing, and interpret your rule. You judge the relationship between them and decide in one way. Then, you later decide that you were wrong, mistaken in your interpretation. That's the nature of recognizing your mistakes. At a later time, something comes to your mind which makes you realize that your judgement was wrong, so you admit that you were wrong, mistaken. Each and every mistake only becomes evident after the fact, and your example is no different. There is no reason to insist that it is impossible to make a mistake in following your own private rule. That's nonsense. Why would a private rule be any different from a public rule in this respect? Is it that in a public case we might have a judge or jury decide whether a mistake was made, rather than the person who acted decide whether a mistake was made? There is no reason why the person who acted cannot decide that a mistake was made, if evidence to that effect is revealed.
You keep saying that it's possible to know whether you have misinterpreted your private rule, or mis-remembered it or maybe correctly interpreted but it in a novel circumstance... But you haven't explained how. How can we distinguish those three things, how could we ever know which it was? How could you later decide you were wrong about your interpretation, what measure of 'right' interpretation do you have by which to make such a judgement?
I told you, evidence comes up at a later time which makes you see that you made a mistake. If anyone else can judge you at a later time, by reviewing the evidence, as having been wrong, why can't you judge yourself as having been wrong, by reviewing the evidence? What's the difference? It's nonsense to think that one cannot judge oneself as having been wrong. We are taught to recognize our mistakes as mistakes, and accept responsibility for them.
Quoting Pseudonym
How can anyone judge someone as having made a wrong interpretation? What measure of "right interpretation" does anyone have? It is the same issue whether the rule is public or private.
You are the one who has been arguing that there is no necessarily "right" interpretation of Wittgenstein. On what principle do you insist that the idea of private rules ought to be rejected because there could be no "right" interpretation? The 'right" interpretation is nothing other than an ideal.
Because you cannot simultaneously hold a rule and faithfully try to interpret it yet make a mistake. We do not have two minds, one with the 'real' rule in it and another trying to understand the what the first one meant by it.
Quoting Metaphysician Undercover
By consensus.
Quoting Metaphysician Undercover
Consensus.
Quoting Metaphysician Undercover
No, because it is impossible to have consensus privately, there's only one of you.
Quoting Metaphysician Undercover
On the principle that a following a rule, and thinking you're following a rule must be two different things, but cannot be privately because we do not have two minds (one with 'the rule' in it and another attempting to interpret it). The 'correct' interpretation of the rule is held publicly, by consensus.
A schizophrenic could hold a rule privately though...???
Quite obviously this is wrong, because it happens all the time that we make such mistakes. It's a matter of misinterpretation. It's not a case of having "two minds", it's a matter of changing one's mind. You make a judgement, apply the rule, then later you realize the judgement was wrong. Have you never changed your mind before?
You just keep making nonsense assertions without thinking about what you are saying.
Quoting Pseudonym
OK, that is your claim, consensus makes "right". I disagree. I see evidence that in many cases when there is consensus, mistake is still made. Therefore it is impossible that consensus makes "right".
Quoting Pseudonym
And this is just more nonsense. Interpretation is what individual minds do. How could "the public" hold an interpretation? If people discuss interpretations of a rule, they use language, and that language must be interpreted by each of them. So each person has one's own interpretation of the "public" interpretation. They could discuss their interpretations of the interpretation, and find consensus, but again, each would have an interpretation of the interpretation of the interpretation. Infinite regress is implied.
This is why Wittgenstein seeks to remove the necessity of interpretation at 201, to avoid this problem. Wittgenstein's claim is not that the correct interpretation of the rule is held publicly. That doesn't make sense. The claim is that the rule itself is what is held publicly. In this scenario, there is supposedly no need for interpretation, either the person follows the rule or not. The problem is that a judgement is implied here, as to whether or not the rule is followed, and there is no "public mind" to make that judgement. We really cannot refer to "consensus" because then we fall into the problem of interpretation.
But how do you realise it was 'wrong'. Different, yes, but 'wrong'?
Quoting Metaphysician Undercover
Great, let's have a look at one of those examples for a public rule then, that might get us somewhere. If you provide an example of a public rule where the 'correct' interpretation or use of it can be derived by some means other than
consensus, we could resolve the problem.
The rest of your argument is based entirely on an error of mine. I meant to say the correct interpretation of the rule is 'judged' publicly, not is 'held' publicly. I can only blame trying to write too fast, I'm sorry to have made you painstakingly explain the infinite regress argument for no reason.
Quoting Metaphysician Undercover
Are you both right in each instance? Just so it doesn't appear as if I'm trolling or insinuating anything, I am inclined to agree with Pseudonym. For, if one were to follow a rule, then the criteria for following it is dictated by something beyond the rule itself.
When you fail to achieve the desired effect for example, you know there was a mistake. If there was a mistake, then something was done wrongly.
Quoting Pseudonym
I am not claiming that "the correct" interpretation can be derived at all. As I said, the right interpretation is an ideal. I don't believe that the ideal is ever achieved. As you argued concerning Wittgenstein's PI. varying interpretations may all be "correct". There is no such thing as "the correct" interpretation, in the sense of the best, the ideal interpretation.
Quoting Pseudonym
This does not avoid the problem I described. Individual human beings make judgements. To say that something is "judged publicly", is to say that there is a vote or some such thing. Just because the majority constitutes consensus, does not mean that the interpretation chosen by the majority is the correct interpretation. And, that interpretation which is accepted by consensus still needs to be itself interpreted by each individual member of that voting public, causing the infinite regress problem.
Quoting Pseudonym
So, you thought at the time that you were following a rule. But you now realize that you were trying to write to fast, and you really weren't following your rule. See what I mean?
Quoting Posty McPostface
Yes, that's exactly my point. That's what allows that an individual can think that oneself is following a rule, a private rule, then later realize that the rule was actually not being followed. Whether or not the rule has been followed, as judged after the action, requires a completely different form of judgement from the judgement of whether or not the rule is being followed, as judged prior to the action. The former is a judgement in relation to the past, how the rule was applied, the latter is a judgement in relation to a future act, applying the rule toward possible acts. One is posterior to the act, the other prior to the act. and these are distinct forms of judgement. One judges actual activity in relation to the rule, the other judges possible activities in relation to the rules. Each has criteria beyond the rule itself, but very different types of criteria.
We're going round in circles and I don't think my replies are helping at all. Let's see if I can clear up a couple of points where I think we might be talking across one another and maybe some idea of what we really disagree on might emerge.
1. Wittgenstein does not conclude anything at all about the solution to the rule-following paradox, a paradox has two sides which together seem incompatible. He does not say which side is right, if anything he's claiming that the paradox emerges because we are confused about the definitions of the terms we use (but that is only a general and speculative conclusion). I'm trying to argue that the paradox is real, I think you might be trying to argue that one side of the paradox exists. If that's the case, then we completely agree, the point is not that one side of the paradox doesn't exist, it's that both sides seem to exist which is a logical impossibility - hence the paradox.
Wittgenstein explains some of this slightly better, I think, in MS109-110.;
"Suppose that a general rule is given. One can, nonetheless, apply the rule only if he understands its application. Suppose, for instance, that someone should translate a sentence from one language into another. He is given the set of sentences to be translated and a dictionary, which is the set of rules of translation:
One could say then: But it is not enough to give him both things; you have also to tell him how to use them as well. But in this way a new plan would be created, which would need an explanation as much as the first one." (MS 109, p. 82).
and
"…I don’t need another model that shows me how /the depiction goes and, therefore/ how the first model has to be used, for otherwise I would need a model to show me the use/application of the second and so on ad infinitum. That is, another model is of no use for me, I have to act at some point without a model." (MS 109, p. 86)
This might be an expression of the paradox that you find more acceptable - It would appear I need a further rule to tell me how to interpret the rule (the dictionary and the foreign language alone does not tell me how to translate), and yet I nonetheless do appear to follow rules without a rule telling me how to do so. Hence a paradox. It is not sufficient to resolve this paradox by showing that we do follow rules without the apparent need for a model to tell us how, that's just one side of the paradox. In order for it not to be a paradox, it is necessary to show the the logic of the first statement is flawed. You're only providing demonstrations of how the second statement appears to be true, but of course the second statement appears to be true, that's why it's a paradox, because the first statement appears to be true also and yet the two contradict one another.
2. The Rule-following paradox is only tangentially linked to the private language argument. The additional difficulty of following a rule private applies to a specific set of rules that are about the correct interpretation of signs. The private language argument is about the correct interpretation of signs, not about the correct application of a prediction. You seem to be likening having a private rule to something like " I wish to stop smoking to make me healthier, I must not have any cigarettes" Such that if a new cigarette-like thing enters the market and you must judge whether smoking it breaks your rule, you can do so by perhaps smoking it, noticing your health is worsening again and thinking "Oh, this must be one of the things I must not have because it is having the effect my rule is trying to avoid"
But this is not the kind of rule Wittgenstein is talking about in the private language argument (though it is a rule that would suffer from the rule-following paradox). So disputing the private language argument and disputing that the rule-following paradox is really a paradox are two different things and you seem to be conflating the two.
To clarify my position, I believe that there is no such paradox. The apparent paradox is the result of Wittgenstein describing "rule" in an unacceptable way. A rule is something which exists within human minds, not in the public domain. What exists in the public domain is physical material, in various shapes and forms, which we all sense, therefore it is public. This includes written, or spoken, physical representations of rules. Wittgenstein fails to properly distinguish between these two, the rule and the physical representation of the rule, when describing what a rule is, so he uses "rule" to refer to both. Since the two are not the same, yet he uses the same word to refer to both, as if they were, an apparent paradox arises. The appearance of a paradox is the result of Wittgenstein using the same word to refer to two distinct things, the rule, and the representation of the rule. That's equivocation.
Quoting Pseudonym
See, his misuse of "rule" is very evident here. "Understanding" the application of a rule is something which occurs within human minds. But then he says that the dictionary "is the set of rules", and this is false. Neither is the dictionary a set of rules, nor is it even a representation of a set of rules. It is a representation of how words are commonly used.
If we proceed and consider a game, and refer to the spoken words, or written words, which represent "the rules of the game", it is evident that these are just physical representations of rules, because it is necessary to interpret these words, to understand "the rules". Common parlance allows us to say that these are "the rules", the written words, but we must differentiate between this common usage, and when we say that someone "understands the rules", because what is understood as "the rules" is the limitations as to how to play the game, not just the words on the paper. What is added by the interpreting mind, is the "ought", and this is not part of the words, it is part of the interpreting mind.
This is the is/ought divide. The words on the paper say something. The player interprets this as how one "ought" to behave. But there is nothing within the words themselves which dictate that the player "ought" to behave in any particular way, the player interprets the words as saying what one ought to do. Since the "ought" is the essential aspect of the rule, making the rule what it is, a guideline for future behaviour, and the "ought" is only assigned to the words by the minds of the writer and interpreter, then what is on the paper does not exist as "rules", in a more strict sense of "rule", when we talk about a person "understanding the rules".
Quoting Pseudonym
The private language argument only follows from this equivocation of "rule". The equivocation of "rule", described above, which Wittgenstein employs, allows him to proceed into the private language argument. If we deny the equivocation, and keep a clear separation between a rule, and a representation of a rule, there is no premise to even start the private language argument. That premise, at 202, concerning the nature of rules, which allows for the private language argument, is rejected. Nor is there the appearance of a paradox. And the appearance of the paradox is what inclines Wittgenstein to propose that premise.