But then something important still usually feels lost in translation when they have to go from abstractions back to words. And what is lost is the clarity gained by abstracting from words to abstractions. — apokrisis

That makes sense. Clarity means determinability. And mathematical and logical propositions are more determinable than empirical truths. The former two are sharpened up when expressed in formal symbolic languages, rather than ordinary informal language.

But philosophy, although it might be informed by formally derived propositions, can be done only in ordinary informal language and much of what it is concerned with is just consists in thinking about the indeterminable as rigorously as possible. Epistemology, metaphysics, aesthetics, ethics; nothing is precisely determinable in any of them, so what "rigorous" means will be only something like "consistent with your (necessary) presuppositions; whatever they might be". The content of our presuppositions is itself never necessary, what is necessary is the (at least provisional) holding of one or the other presupposition, in order to get started.

In ordinary analysis the continuum R is connected in the sense that it cannot be split into two non empty subsets neither of which contains a limit point of the other. In smooth infinitesimal analysis it has the vastly stronger property of indecomposability: it cannot be split in any way whatsoever into two disjoint nonempty subsets. — SEP article, Continuity and Infinitesimal

I think this passage is explaining what I was trying to say... that if you have a cake for which the law of excluded middle fails, you can't just slice the cake.

I think that's the fundamental problem here, with Deleuze, and with the aesthetic approach to epistemology in general. Insofar as it purports to be a sub-representational account of thought, it cannot be represented - it literally cannot be thought or talked about. — Aaron R

The issue is more subtle than this, although I admit that in my haste to distinguish intensive (analog) differences from the Kantian 'thing-in-itself' I moved too quickly. First, sub-representational intensities are meant to account for extensional magnitudes: in Deleuze's words, they are the sufficient reason of all phenomena, "the condition of that which appears". Second - and here is where I moved too fast, we don't know intensity 'directly', but rather "we know intensity only as already developed within extensity, and as covered over by qualities."

As far as logical status of intensity goes, intensity thus occupies an undecidable place within any system of representation: it can only be known through representation, but is is nonetheless not of the representational register. It's status is strictly correlative to that of the digital cut itself, which neither belongs to the system of representation nor is merely external to it. Hence the paradoxical status of intensity with respect to the question of knowledge: "[Intensity] has the paradoxical character of the limit ... [It] is both the imperceptible and that which can only be sensed."

Here is where things get complicated, but I'll try and do my best to explicate the ideas. If you recall that what's at stake is a 'critique of pure logic', then the idea is to introduce 'extra-formal’/‘real' constraints on the the exercise of what might otherwise be purely syntactic logical manipulations which might simply follow transitively from an established set of axioms. For Deleuze, intensive differences are precisely what force 'real life' (extra-formal) constraints of 'existence' on logic, making logic no longer a formal and arbitrary play of symbolic manipulation, but beholden to a specific existential situation, as it were.

Thought - which just is representational - must be ‘forced' to think under the aegis of what Deleuze refers to as an ‘encounter’ with sub-representational intensities which impose 'real constraints' on thought. These constraints shift the modality of thought from the order of the arbitrary to the necessary: "if necessity is only ever the necessity of an encounter, and of a relation that this encounter gives rise to within us, a relation whose nature cannot be known prior to the forced movement it induces, then we must reconsider the meaning of the arbitrary. The concern of critical philosophy cannot be bound up with evaluating truth from a position of relative or extrinsic indifference … When truths are separated from the necessity of an encounter they become abstract, which is to say, they are reduced to being merely possible or hypothetical.” (Kieran Aarons, The Involuntarist Image of Thought).

There’s a lot more to say here. I’ve not really given a full blown account of intensive differences, nor the manner in which they force us to think, so much as focused on attempting to answering the charge that buying into the notion plunges us into the myth of the given. At most, I’ve focused on the status of intensive differences with respect to thought, but I’ve already gone on too long. By all means ask any follow up questions though, cause these are bloody good thought-encounters for me. But I don’t want to prattle on too long. In the meantime, I'd direct you to this paper by Peter Kugler which takes up exactly how to make sense of the above using Ryle's notion of categories. Will probably elaborate in a next post if you want.

I think this passage is explaining what I was trying to say... that if you have a cake for which the law of excluded middle fails, you can't just slice the cake. — Mongrel

No, but you can make the law of the excluded middle apply by imposing a rule which would, on that basis, arbitrarily split said cake. That’s the whole point of digitisation: you take something that cannot be ‘naturally’ split, then you arbitrarily define a rule by which to impose a distinction on said continuum, then you use that rule to split the continuum from the ‘outside'. Of course, this rule will always leave a remainder, in the form of self-referential paradoxes. In math, this rule is the empty set, and it’s corollary, zero.

This is how you do it: Take a set, S. Then, you find the compliment of S, which just so happens to be the empty set, ∅ (S-S = ∅). Now that you’ve done this, you’re in a great position because the empty set plays a double role. Not only is it the compliment of S, it is also a subset of S, to the extent that every set contains the empty set. Note that the empty set is thus is both ‘inside’ and ‘outside of S, occupying exactly the paradoxical place which we said a rule for distinction would occupy.

Having done this, you can generate the entirety of the number line by asking how many elements belong to the empty set (=1), and then recursively asking how many elements belong to that set and so on ad infinitum. Ta da. You’ve now digitised the continuum.

The problem, of course, as with any digitisation, is whether or not 0 belongs to it. The answer is strictly undecidable. Wilden: "zero is not simply a number as such, but a rule for a relation between integers… zero is implicitly defined as a meta-integer, and indeed its definition is what provides the RULE for the series of integers which follow it.” Zero, like negation, is a higher-order, reflexive rule about the continuum on the basis of which we can divide it, provided we cannot situate either negation nor zero properly in that continuum itself.

Which is exactly the point.

I'm too tired to reply to your mammoth post properly tonight, but I wanted to quickly post that it seems to me our differences come down to whether or not one accepts or rejects pansemiosis. I reject it. Semiosis only makes sense at the level of life - so I have no problem with biosemiotics - but I think your attempt to extend those principles beyond that 'border' are illegitimate and not well-founded. In lieu of a proper reply, you might be familiar with the hyperlinked conversation between Salthe and others over exactly this question. Needless to say, I don't take Salthe's side here.

No, but you can make the law of the excluded middle apply by imposing a rule which would, on that basis, arbitrarily split said cake. — StreetlightX

"And that's how we make the golf ball go into the hole!" -Zeno

That's my Zeno impression. Later, I'll do my Aristotle explains what all of this has to do with God.

This is how you do it: Take a set, S. Then, you find the compliment of S, which just so happens to be the empty set, ∅ (S-S = ∅). Now that you’ve done this, you’re in a great position because the empty set plays a double role. Not only is it the compliment of S, it is also a subset of S, to the extent that every set contains the empty set. Note that the empty set is thus is both ‘inside’ and ‘outside of S, occupying exactly the paradoxical place which we said a rule for distinction would occupy.

Having done this, you can generate the entirety of the number line by asking how many elements belong to the empty set (=1), and then recursively asking how many elements belong to that set and so on ad infinitum. Ta da. You’ve now digitised the continuum. — StreetlightX

It appears that when you asked how many elements belong to the empty set and came up with one, you were already thinking in discrete terms.

Zero, like negation, is a higher-order, reflexive rule about the continuum on the basis of which we can divide it, provided we cannot situate either negation nor zero properly in that continuum itself. — StreetlightX

Doesn't strike me as intuitional to say that Zero is a higher-order, reflexive rule about the continuum on the basis of which we can divide it. In fact it makes close to Zero sense to me. Um... what's the basis for this rule, then?

It appears that when you asked how many elements belong to the empty set and came up with one, you were already thinking in discrete terms. — Mongrel

Yes, and? A more fun way to understand the whole deal with the empty set is that it's like distinguishing the cake from the not-cake, which means that you take the continuum as such as a discrete element. But of course, there is no not-cake 'in' cake. And then you work your way from there.

Doesn't strike me as intuitional to say that Zero is a higher-order, reflexive rule about the continuum on the basis of which we can divide it. In fact it makes close to Zero sense to me. — Mongrel

Take it up with math, not with me.

Um... what's the basis for this rule, then? — Mongrel

The basis is always methodological. What are you trying to achieve with your rule?

Take it up with math, not with me. — StreetlightX

You can't take it up with math. You can ask a phil-o-math person... which would obviously be Nagase, but he's busy writing some thesis.

To review.. the criticism that was brought was:

That the digital doesn't just fit "loosely" on the analog. I think you're agreeing with that. You note that we use Zero to create a sliced cake... apparently using the Unslice-able Cake as the primal One.

I've found my interests coming back around to Leibniz lately. This is the second time.

No - we 'create' zero by digitizing the cake and the not-cake. And not-cake is - as the name implies - a reflexive operation: it is a meta-statement about cake.

I might put it to my fellow mods that all discussion take place in terms of cake now btw.

You're saying the primal distinction is between the cake and the not-cake.

You can't digitize the Continuum Cake. It has a high glue content and it just stretches forever if you try to take a piece out of it.

You can bake an entirely different cake that can be sliced because it's fundamentally atomic to begin with.

Not if I distinguish between cake and not-cake. This allows me to create a digital model of the cake which I can work on in all it's discrete glory.

Digital model. Hmm.

Different tack: I think we can discover whether the categories we're talking about are apriori or aposteriori using the Locke/Hume/Kant trick of asking about what is and isn't imaginable.

Can you imagine a cake that can't be sliced? I say no. Therefore, digital is apriori.

Can you imagine that space itself is atomic? I say no. Therefore, analog is apriori.

They both are. They're conditions of knowledge of the cake.

Not much of a trick.

You can add in a somersault if you want.

This is how you do it: Take a set, S. Then, you find the compliment of S, which just so happens to be the empty set, ∅ (S-S = ∅). Now that you’ve done this, you’re in a great position because the empty set plays a double role. Not only is it the compliment of S, it is also a subset of S, to the extent that every set contains the empty set. Note that the empty set is thus is both ‘inside’ and ‘outside of S, occupying exactly the paradoxical place which we said a rule for distinction would occupy.

Having done this, you can generate the entirety of the number line by asking how many elements belong to the empty set (=1), and then recursively asking how many elements belong to that set and so on ad infinitum. Ta da. You’ve now digitised the continuum. — StreetlightX

Not quite sure what's supposed to be going on here, but whatever it is, you can't generate the Reals that way.

Here is where things get complicated, but I'll try and do my best to explicate the ideas. If you recall that what's at stake is a 'critique of pure logic', then the idea is to introduce 'extra-formal’/‘real' constraints on the the exercise of what might otherwise be purely syntactic logical manipulations which might simply follow transitively from an established set of axioms. For Deleuze, intensive differences are precisely what force 'real life' (extra-formal) constraints of 'existence' on logic, making logic no longer a formal and arbitrary play of symbolic manipulation, but beholden to a specific existential situation, as it were. — StreetlightX

I would like to suggest that what you're calling "the critique of pure logic" really boils down this: pointing out that classical accounts of represenatation do not (and cannot) account for the processes of concept creation and/or concept revision (e.g. "creative problem solving", "learning", ect.). The failure of classical logic in this regard is in turn grounded a failure to deal with what you have called (via Deleuze) "the encounter" - which is the event in which some schema of representation is forced to change through confrontation with "the world" via sensation and perception. The encounter confronts us with "the problem", prompting the revision of representation that is "the solution" to the problem.

So in Deleuze we have this thread of ontological duality running through his philosophy, and manifesting in the interrelated dichotomies of problem and solution, sub-representation and representation, intensive and extensive, and (ultimately) virtual and actual. On this account, what we call "experience" is just what happens in the "in between" space of the interminable systole and diastole (i.e. "the eternal return") of the dynamics of these mutually immanent "poles" of reality.

I'm on board with the critique as far as it goes, but am not so sure about the alternative being provided. Again, one has to wonder as to the epistemological status of the virtual given that it must necessarily remain "papered over" by representation. If we subtract out the contents of our representations of the virtual, what's left? The noumena? The shadowy realm about which we can know nothing more than that it causally constrains our representation of it?

Personally, I prefer the Perceian strategy for dealing with the noumena by interpreting it in primarily epistemological rather than ontological terms. So instead of being the shadowy, causal underbelly of the world, it is transposed into the content of the ideal limit of inquiry (e.g. regulative rather than constitutive). It is still a limit concept, but it no longer entices us towards the intellectual bankruptcy of mysticism, and instead pushes us toward the satisfaction of the insatiable desire to know. On this view, the world that causally constrains thought is just the world as we have come represent it so far (what else could it be?) - that is, the world as described by science and (where science fails) common sense. This position is not free from problems, but what position is?

And in that vein, I'll also state that the more I engage in these kinds of discussions the more it seems that the selection of one's metaphysics and epistemology reduces to a matter of personal taste and temperament. Every metaphysics/epistemology has it's strengths and weaknesses. None is immune from the confrontation of certain vexing problems that seem to be inherent within the structure of thought itself. I'm rambling now, so I am going to end on that slightly pessimistic note.

it seems to me our differences come down to whether or not one accepts or rejects pansemiosis. — StreetlightX

If you could demonstrate that you understood it, your rejection would be a lot more convincing.

Don't forget that I've always said biosemiosis is definitely something new in nature - the development of full-blown digital-strength symbolism to allow for autonomous systems within the Universe.

Physiosemiosis would be vaguer in just being analogic or iconic. And it would be even these in a vaguer sense as the interpreter is "the Universe" as a system - a material system without yet any symbol systems operating within it in their autonomous (not-A) fashion.

So pansemiosis - as Salthe works to define it - is simply the assertion that the Universe is self-organising and comes into existence as a global regulative habit. It is a view rooted in dissipative structure theory and far-from-equilibrium thermodynamics. Natural law is like the self-closure that is the eruption of constraining convection currents in a Benard Cell. Semiosis speaks to the formation of the negentropy or memory by which a Universe becomes its own vehicle for a generalised production of entropy.

Thus it seems in all these ways precisely a thesis that you would agree with. You would have to explain to me how it says something different in your view.

Remember also that the new thing is that the biophysics of the nanoscale has now empirically identified the physical point where a transition from physiosemiosis to biosemiosis can happen - or indeed, is inevitable. I wrote that up in this thread - http://forums.philosophyforums.com/threads/the-biophysics-of-substance-70736.html

So now we have identified a convergence point where material being has a critical instability - an edge of chaos cusp of order~disorder - that allows "digitality" in a physically real sense.

A problem with your highly abstracted exposition is that you make a huge mystery of how the digital cut can be imposed on the analog world - the slice that cuts the cake. Somehow the cake breaks apart as intended without your knife physically doing anything - waving it wishfully or threateningly suffices.

Your use of Wilden's computer analogy encourages this. A computer has just this kind of symbolic disconnection from the world. The software is granted the security of utterly stable hardware and so doesn't have to think anything about its operation. Whereas with life, and semiosis generally, the situation is the precise opposite. It is all about the regulation of a fundamental instability, a fundamental vagueness. And the more on the cusp of the edge of chaos things are, the greater also the semiotic range of regulative possibilities. (Have you ever read Scott Kelso for example? - https://mitpress.mit.edu/books/dynamic-patterns)

So despite the fact you normally claim to be an enactivist, in this thread you have argued from the basis of representationalism - analog and digital computation both being ways to represent the world. And so any digital cut remains virtual rather than actual. The computer can click and whir away doing its digital or analog thing and it makes no bloody difference to the world unless somebody - usually a human - takes notice of its syntactical mapping and treats it as a sign of something about the world.

Semiosis - like enactivism - says that is ridiculous. The digital cut has to be a real cut out in the world. The cake must be sliced - or at least nudged just enough for it to reorganise itself into two parts because it was on the cusp of just such an entropic bifurcation.

So this is the mystery that semiosis solves.

The regular mechanical way of looking at the world presumes that the ground of any hierarchical complexity must be rock solid stable. You have to have something crisp and definite - like atoms - to begin any construction work. The cake is there and is never going to cut itself because cakes have had any such dynamism or self-organisation baked out of them. And all that makes it a real material mystery how any amount of symbolic activity - analog or digital computation - is going to make a difference. The cutting can be imagined, yet where is the power to execute?

But the self-organising semiotic view of the world says instead that you get these major transition zones due to criticality. Now reality is as unstable as it can be - suspended between two states. And the slightest nudge can tip it in either direction. So there is a digitality inherent in the material state (it can distinctly go in either direction just due to spontaneous fluctuations). And then that digitality can be made extrinisic by a symbol system which retains only the slightest physical presence in that world. A system of signs can compute where and when cake self-cutting should happen. Then deliver the almost infinitesimal physical nudge that tips the balance.

So first the physical world does its bit by presenting the potential - some point of absolutely poised instability. And then a minimal bit of physical machinery - a nudging mechanism controlled by as much background symbolic computation as you like - can exploit that eminently controllable situation.

Thus the digital cut imposed in recursive fashion via a negative mark (a pointing towards whatever state a biifurcation happens not to be in) is no longer the kind of virtual phantom act it must be in your framing of things, it now has an actual physicality. It has a size. Indeed it has the particular universal scale now discovered by biophysics.

So I know you think you reject pansemiosis, Salthe, vagueness, and indeed anything that I might mention that you are not already familiar with. But really you are just in the process of getting there.

And one of the presumptions you might not realise you have been making is that existence must be founded in the stable, when the whole point of any view founded on process thinking - such as enactivism - is that it is instability which makes the very idea of regulation possible in the world.

This is how you do it: Take a set, S. Then, you find the compliment of S, which just so happens to be the empty set, ∅ (S-S = ∅). Now that you’ve done this, you’re in a great position because the empty set plays a double role. Not only is it the compliment of S, it is also a subset of S, to the extent that every set contains the empty set. Note that the empty set is thus is both ‘inside’ and ‘outside of S, occupying exactly the paradoxical place which we said a rule for distinction would occupy. — StreetlightX

I don't know if you're using math metaphorically here, but the compliment of S is going to be relative to some set X. If X = S, then, yeah, the empty set is its complement. To say that the empty set is both inside and outside of S is a bit of cheat. It's a subset of S but not an element of S (in general). I don't doubt that your getting at something interesting about rules for distinction, though.

Having done this, you can generate the entirety of the number line by asking how many elements belong to the empty set (=1), and then recursively asking how many elements belong to that set and so on ad infinitum. Ta da. You’ve now digitised the continuum. — StreetlightX

Are you saying the empty set contains an element? (It definitely doesn't.) You then mention the continuum, but R is typically constructed with subsets or Q (cuts), etc. (Maybe you know all of this.) The measure of the computable reals is 0, so it's not such an intuitively satisfying "digitalization." But maybe this is all metaphorical. If so, perhaps that should be stressed.

The problem, of course, as with any digitisation, is whether or not 0 belongs to it. The answer is strictly undecidable. Wilden: "zero is not simply a number as such, but a rule for a relation between integers… zero is implicitly defined as a meta-integer, and indeed its definition is what provides the RULE for the series of integers which follow it.” Zero, like negation, is a higher-order, reflexive rule about the continuum on the basis of which we can divide it, provided we cannot situate either negation nor zero properly in that continuum itself. — StreetlightX

So this is from System and Structure. It looks like good reading, but it doesn't seem that we are in Kansas anymore (actual math), but instead in a realm where Freud is relevant and "presence and absence fill a continuum." That's cool. Just sayin'.

I don't know if you're using math metaphorically here, but the compliment of S is going to be relative to some set X. If X = S, then, yeah, the empty set is its complement. To say that the empty set is both inside and outside of S is a bit of cheat. It's a subset of S but not an element of S (in general). I don't doubt that your getting at something interesting about rules for distinction, though. — Hoo

Set theory probably has the problem that it builds in the distinction SX hopes to derive. It's weakness is that its brackets that bound possibility are themselves so definite and unexplained as features of the world.

But there would be two ways of looking at this.

Either the brackets - {....} - exist in deus ex machina fashion as if someone outside constructed boundaries large enough to contain anything, and thus both everything and nothing (as the crisp complementary limits on vague anythingness!).

Or instead the brackets in fact just represent the simpler thing of being the emergent complementary limits on such a naked state of possibility. The brackets stand for the fact that possibility has its own inherent limits. In saying something is possible, everything and nothing, infinity and zero, already also exist in negative recursive fashion as now the places where everythingness and nothingness put a stop to somethingness.

So - and here is the difficult bit - the limits on being are precisely that which doesn't itself exist. A boundary is where reality stops. And so the boundary itself is unreal or non-existent - even if it seems to have brute causal presence in being "a limit".

This is why I objected to SX's idea of boundaries as something like a 1D line drawn across nature - a single dimensionless feature that somehow bisects reality to make it binary.

Instead - organically - the metaphysical-level logic is that of the dichotomy. The self-organisation that results in a system arising within its own opposing boundaries or limits. The crisp brackets of the set are formed as a result of the action arising within them. The contents are producing their own container - so as to be now definitely "the contents" rather than just vaguely that.

This would be why folk feel that category theory is a better foundation for maths than set theory. It has that embedded dichotomistic view in the mutually exclusive/jointly exhaustive formulation of "structure and morphism". Instead of the container and contents metaphor, we have a organic distinction of constraints and freedoms, organisation and change.

So set theory could be naturalised by recognising the opposed brackets as standing for complementary poles of being - the opposed limits you need to arrive at to have the third thing of the individuated something that can now stand between.

It is then a further thing to give a name to these limits - to call them out as it were, even though they are by definition precisely what does not exist (even as possibility!). So we can speak about infinity, we can speak about zero, as concrete real things. Just as we can talk about all the metaphysical-strength limit states like the discrete~continuous, vague~crisp, stasis~flux, matter~symbol, chance~necessity, part~whole, atom~void, etc, as being real in their limit state unreality.

And that is very powerful from a modelling or reality-mapping point of view. Just look at 2500 years of Western intellectual history. But it also makes us prone to the fallacy of misplaced concreteness that the process view warns us of.

One last point on SX's idea of boundaries as just lines, he would do better to consider Spencer-Brown's diagramatic use of circles as the simplest shapes to form an inside vs an outside - a canonical act of digital symmetry-breaking. Or even better still, go further back to the source of those laws of form in Peirce's own diagramatic re-formulation of logic.

http://mentalmodels.princeton.edu/papers/2002peirce.pdf

http://homepages.math.uic.edu/~kauffman/Peirce.pdf

Set theory probably has the problem that it builds in the distinction SX hopes to derive. It's weakness is that its brackets that bound possibility are themselves so definite and unexplained as features of the world. — apokrisis

As I see it, math is machine-like. "Here are formal definitions. Here are rules of inference. See how these definitions are related in terms of those rules of inference." The formal definitions tend to have intuitive appeal of course, but we're aren't allowed to use intuition directly. The ghost of intuition must be incarnated in the symbolism.

As I see it, the formal definition of "set" tries to capture the intuition of "gathering up into a unity." All things as things are unities. The tail and the nose and the fur and so on have been gathered up as the dog, for instance. It's as there is always already a logical circle drawn around any particular thing, perhaps giving it its thing-hood, cutting it out from the background automatically. But then sets are also (intuitively) the extension of properties, which surely inspired the axiom of extensionality.

So set theory could be naturalised by recognising the opposed brackets as standing for complementary poles of being - the opposed limits you need to arrive at to have the third thing of the individuated something that can now stand between. — apokrisis

I think we get this from writing R as (-inf, inf).

I can't respond to much of your post, since I don't have a feel for it. But I have checked out a book on Peirce, so maybe I'll understand you better after I read it.

As I see it, math is machine-like. "Here are formal definitions. Here are rules of inference. See how these definitions are related in terms of those rules of inference." The formal definitions tend to have intuitive appeal of course, but we're aren't allowed to use intuition directly. The ghost of intuition must be incarnated in the symbolism. — Hoo

That's right. Once you have axioms, you are good to go with the deductions. It all unfolds mechanically in a predestined fashion.

But what is the meta-theory about forming axioms - the semantic residue animating the unfolding syntax?

I would argue that it is dialectic or dichotomistic metaphysics. That is what presents us with our "binary" choices. We can posit the axiom of continuity - having identified it as one of two choices. Reality could be fundamentally discrete or continuous. Well, let's pick continuous for the sake of argument and run with that, see where it leads.

As I see it, the formal definition of "set" tries to capture the intuition of "gathering up into a unity." All things as things are unities. The tail and the nose and the fur and so on have been gathered up as the dog, for instance. It's as there is always already a logical circle drawn around any particular thing, perhaps giving it its thing-hood, cutting it out from the background automatically. But then sets are also (intuitively) the extension of properties, which surely inspired the axiom of extensionality. — Hoo

Well the relevant axiom is the axiom of choice. It starts by presuming individuated (crisp and not vague) things, events, properties, whatever. And given that is the case, forming collections becomes trivial in being trivially additive and subtractive. One can construct any unity (or deconstruct it to leave behind "nothing").

I think we get this from writing R as (-inf, inf). — Hoo

Or I would prefer to think of it in terms of the reciprocal limits defined by the notions of the infinite vs the infinitesimal. This is the strictest way of defining each limit on possibility in terms of its other.

Positive and negative infinity are hardly marking bounds in claiming to point in either direction in terms of the unlimited.

I would argue that it is dialectic or dichotomistic metaphysics. That is what presents us with our "binary" choices. We can posit the axiom of continuity - having identified it as one of two choices. Reality could be fundamentally discrete or continuous. Well, let's pick continuous for the sake of argument and run with that, see where it leads. — apokrisis

I like dialectic. That's the process. The thesis swells (via anti-thesis then synthesis, repeat) and becomes more capable.
I like the instrumentalist approach. It's not about what's behind the manifest image. It's about what we can do within the manifest image using our theories. Think "prediction machines" or "manipulation machines." It's probably natural for the scientist to think in terms of representation, as with a mathematical sense of "X-ray vision" that pierces through the manifest image. It would also be hard to do math as a sincere formalist. One wants to prove something about objects that exist inter-subjectively. So there are atoms and real numbers, but this "are" doesn't seem absolute. It flickers in the context of purpose and focus. How would we cash out reality as continuous? How would it be established? Our most predictive/manipulative theory based on the real numbers? Or on geometric intuition of flow?

Well the relevant axiom is the axiom of choice. It starts by presuming individuated (crisp and not vague) things, events, properties, whatever. And given that is the case, forming collections becomes trivial in being trivially additive and subtractive. One can construct any unity (or deconstruct it to leave behind "nothing"). — apokrisis

The AC is often stated as the existence of a choice function. Are you sure you don't have another axiom in mind? I think the logical use of equality keeps things distinct in math generally, not just in set theory. We simply have x = y or not (x = y). All of x's properties are "naked" if we have the eyes to see it. Of course complicated deductions are not obvious, so some properties are invisible, although "already there" in some sense. (The relationship of time and classical logic is probably quite deep. )

The AC is often stated as the existence of a choice function. Are you sure you don't have another axiom in mind? I think the logical use of equality keeps things distinct in math generally, not just in set theory. — Hoo

What I have in mind is the assumption that you can just pick out individuals and throw them into different contexts freely. But what if that identity was contextual? It's like imagining being able to scoop a whorl of turbulence out a river with your bucket. So the AC shows that kind of assumption at work. But then all of maths pretty much assumes that.

How would it be established? Our most predictive/manipulative theory based on the real numbers? Or on geometric intuition of flow? — Hoo

Geometry always beats algebra for me. But note Michael Atiyah's view that the two are dichotomous and reciprocal. Geometry is manipulation in space and algebra in time. And anything describable in the one reference frame can usually be flipped over into the other, as with symmetry groups or Cartesian curves. So dialectics or duality applies right at the heart of mathematical development.

See:

https://web.math.rochester.edu/people/faculty/cmlr/Advice-Files/Atiyah-Mathematics.pdf

That's right. Once you have axioms, you are good to go with the deductions. It all unfolds mechanically in a predestined fashion.

But what is the meta-theory about forming axioms - the semantic residue animating the unfolding syntax? — apokrisis

There is an important issue here. In philosophy, an axiom is a self-evident truth. In mathematics, an axiom may be anything which does not contradict the mathematical system which it is put to use within. So in philosophy, an axiom is necessarily true, while in mathematics, an axiom is a logical possibility.

We can posit the axiom of continuity - having identified it as one of two choices. Reality could be fundamentally discrete or continuous. Well, let's pick continuous for the sake of argument and run with that, see where it leads. — apokrisis

So, as a philosophical axiom, we cannot just pick any axiom, it must be self-evident. We have evidence that objects are bounded, and "object" may be defined in such a way that an object is necessarily bounded, so we could pick an axiom such as "objects are bounded".

With respect to continuity though, as I stated earlier in the thread, that some aspect of reality is continuous, is implied through observations of reality, and inductive reason. Since it is implied, that some aspect of reality is continuous, this is not self-evident, we cannot pick continuity as an axiom. The assumption of continuity must be justified.

So, as a philosophical axiom, we cannot just pick any axiom, it must be self-evident. We have evidence that objects are bounded, and "object" may be defined in such a way that an object is necessarily bounded, so we could pick an axiom such as "objects are bounded".

With respect to continuity though, as I stated earlier in the thread, that some aspect of reality is continuous, is implied through observations of reality, and inductive reason. Since it is implied, that some aspect of reality is continuous, this is not self-evident, we cannot pick continuity as an axiom. The assumption of continuity must be justified. — Metaphysician Undercover

Well self-evident is always going to be a suspect claim.

But anyway, are bounds not self-evidently continuous? So if there are (discrete) objects, then continuity is also an aspect of your axiom of object boundedness?

But anyway, are bounds not self-evidently continuous? So if there are (discrete) objects, then continuity is also an aspect of your axiom of object boundedness? — apokrisis

I don't think it is self-evident that boundaries are continuous. A dotted line makes a non-continuous boundary. I think the best example of boundaries that nature gives us, is the boundary of a physical object, which we see with the visual sense, and touch, feeling it with the hand or other body part. The texture of those boundaries indicates that they may not be as continuous as they appear to be. Of course the science of chemistry indicates to us that the boundaries between substances cannot be considered to be continuous at all.

If we deny the reality of these boundaries, saying that the boundaries of physical objects are not real boundaries at all, what are we actually doing with this denial? We are denying the actual examples of boundaries, in favour of an ideal boundary. We simply assume that boundaries are continuous, as a mathematical type of axiom, an ideal which has not been justified. Then the boundaries which are shown to us do not fulfill the qualifications of the ideal, so we deny that they are boundaries. Now the ideal boundary must be justified as a true example, or it should be dismissed as not properly representing the boundaries which we know of.

That is why I suggested earlier in the thread, that we consider the boundary between future and past, in time. Perhaps this boundary can justify the ideal continuous boundary which you desire as an axiom.

A dotted line makes a non-continuous boundary. — Metaphysician Undercover

Wouldn't it be leaky or .... vague?

We simply assume that boundaries are continuous, as a mathematical type of axiom, an ideal which has not been justified. Then the boundaries which are shown to us do not fulfill the qualifications of the ideal, so we deny that they are boundaries. Now the ideal boundary must be justified as a true example, or it should be dismissed as not properly representing the boundaries which we know of. — Metaphysician Undercover

That just puts us back dealing with dichotomies as I routinely argue. We can have the ideal or axiomatic notion of a continuous boundary because we also have the ideal/axiomatic notion of what would be the most leaky possible boundary - one that is discrete instead of continuous, all holes and no bounds like a sieve.

So we have two true notions - the unbroken and the broken. And we can then measure anything in the real world by how close or far it is from those bounding ideal limits.

Dotted lines of course usually mean "tear here" so they are suggestions left for you to complete. The would be exactly halfway between unbroken and broken in that sense.

What I have in mind is the assumption that you can just pick out individuals and throw them into different contexts freely. But what if that identity was contextual? — apokrisis

Oh, then we're on the same page. Math is charming because it escapes this mess by fiat. But away from math identity is tangled in context. "No finite thing has genuine being." (Hegel). The concrete reality (the complete reality) is singular. To understand a blade of grass fully is to understand the totality itself. Essences are describable as nodes on a network, utterly interdependent. The same thing applies to sentences. Meaning appears to me to be radically holistic. We would like it to be more atomistic so that we could normalize metaphysics. We can't get the metaphysical/philosophical axioms that MU mentions because meanings are context dependent. That's the temptation of taking propositions modulo actions. If two different strings of marks and noises function the same way as rules for actions, they are equivalent. This is just a normative rule of thumb. But this is why I'm not excited about metaphysical issues as they become distant from values or useful "framing" metaphors. The "language is a tool" metaphor (as opposed to "language is a mirror") is basically for me anyway the essence of pragmatism. We don't ask if a tool represents accurately. We see if it does what we want done. As we are fairly certain of our desires, it offers a streamlined epistemology.

You are contradicting your proposed axiom though. The axiom was that boundaries are continuous. I objected, saying that this is not self-evident. How does proposing two types of boundaries, continuous and non-continuous, help to solve the issue?

We can't get the metaphysical/philosophical axioms that MU mentions because meanings are context dependent. — Hoo

So let me see if I have this straight, the position you're arguing. It is useless to seek self-evident axioms, as there is no such thing, because meaning is context dependent. Therefore we should only use mathematical axioms, as apokrisis suggests, which have crisply defined, and fixed meaning within a mathematical system. This entails that anything which is logically possible is also true.

You are contradicting your proposed axiom though. The axiom was that boundaries are continuous. I objected, saying that this is not self-evident. How does proposing two types of boundaries, continuous and non-continuous, help to solve the issue? — Metaphysician Undercover

Sorry, I thought it was the axiom you had proposed. But instead your self-evident axiom is that objects are bounded.

So still my answer would be the same. Metaphysical-strength axioms seem self-evident when they result from dichotomous reasoning. If a pair of possibilities are mutually exclusive and jointly exhaustive, then in being the mutual limits on such possibility, and in exhausting all other possibilities, they would have the status of necessity.

And that has long been accepted of the continuous~discrete. Together they are as far as you could go in making a contrast between the connected and the disconnected, the integrated and the differentiated, the related and the isolated, etc.

But then as I say, my own take is that dichotomies only do produce ideal limits. And limits are boundaries in marking where reality ceases to be some thing. Which in the metaphysical case, is where reality ceases itself to exist. And so while reality might approach the ideal of either the discrete or continuous with asymptotic closeness, it can never actually arrive exactly there because the boundaries are not part of existence. They mark (in our minds) the limit, so the exact point where the business of existing has halted.

So now we could talk the same way about your own proposed dichotomy here - objects and boundaries. You can see how it is actually parasitic on the continuous~discrete as a metaphysical axiom. We can imagine the discrete, individuated, differentiated, isolated thing which is an object because we can imagine the complementary thing of it having a continuous, unbroken, integrated, related boundary - a boundary which is a global limit on the object in marking the point where all its discrete being suddenly stops.

So yes. The idea of a bounded object seems pretty convincing. But boundaries in reality are often pretty vague. Or if crisp, designed in fact to be leaky.

Any river or coastline is a pretty vague boundary. Tides and floods shift the margin between water and land continually. Tracing a river to its source in some clutter of springs and tributaries is always a contentious affair.

On the other hand, country borders, cell membranes, and other semiotic lines drawn across the world, are not just leaky, they are designed to be porous - porous in a way that is regulated. A border or membrane is a boundary which has to have holes so as to allow the object - the nation or organism - to make the right kind of material transactions to continue to persist as the kind of objects that they are.

So the idea of a bounded object is a crisp metaphysical ideal that in reality only really exists in this fashion.

Even a rock has vague bounds as an object. It is always subject to erosion. And at what point exactly - with metaphysical-strength or Platonic perfection - is some silicon or iron atom crossing the boundary from being part of the solid rock to part of its history of eroded material? Or is the mud on the rock, part of the rock as "an object"? If not, why not?

And then where an object in fact has the power to self-define its own boundaries (when it is an organism), or when it is an artifact (like a nation or a plastic cup) where it is us who imposes some idea of a definition, then really any boundary is a constraint imposed on material vagueness. It is regulation of erosive or dissipative processes designed to reconstruct what the world would generally aim to deconstruct over time.

So on the one hand, we can easily imagine a world of bounded objects. We can axiomatise a metaphysical dichotomy in that fashion - one that is built up from ancient debates about the continuous and the discrete, the one and the many, to arrive at an atomistic conception of bounded objects.

But then when that axiomatised conception is put to the emprical test, we find that reality is different. It has a further developmental dimension to it. Reality is founded more on flux than stasis. The Universe is one vast sea of erosion. And now - metaphysically - its ultimate other must be the counter-move of regulative habit. Boundaries are really constraints on dissipative freedom - or vagueness. Boundaries are the semiotic information that form up stable object-ness in a fundamentally unstable world.