I thought you'd be interested on Terence Tao's thoughts on the development of mathematical skill. — fdrake

Thanks very much for that. I think this idea, suitably adapted, might dovetail nicely with Wiggins's account of someone's conception of a concept, construed as the Fregean sense of this concept (while the concept itself still lives at the level of Bedeutung, or of Fregean reference). Tao's idea of the development in skill/understanding lines up with Wiggins's idea of the improvement of a conception that enables, at once, a better grasp of the sense of a concept and an active participation into its constitutive practice.

Wiggins develops this idea most fully in The Sense and Reference of Predicates: A Running Repair to Frege's Doctrine and a Plea for the Copula

Just like the pragmatized (embodied and situated) neo-Kantian account of Sellars, Haugeland and Bitbol, Wiggins's account of the way in which we grasp concepts steers a middle path between anchoring them into merely contingent features of the embodied subject or making them fully 'independent' of us (in the manner of modern Platonism). The precise account of this cognitive anchoring, though, appeals to some features of modality and reference that are indebted to Frege and to Kripke (and Putnam). Those features have been highlighted by Gareth Evans, also, in The Varieties of Reference (in the chapter on proper names, which discusses reference to natural kinds, also). I've written some posts about this many years ago on a Yahoo discussion group. I'll try to locate them.(*)

(*) Here they are: see mainly this post, which was a followup on this one.

I am interested in reading more Haugeland, though, (as soon I discover where his essays are stashed.) — frank

When we discussed Pattern and Being, it was available online. Truth and Rule Following (which further develops the same ideas) only has been published as the last chapter of Having Thought: Essays in the Metaphysics of Mind, HUP, 2000. I can't legally post pdf documents but I can PM you a treasure map.

I'd be really interested in any more you have to say about that. — frank

If you don't mind me referring you back to old posts of mine, I've sketched my understanding of the significance of synthetic a priori propositions, here and there, with reference to John Haugeland's neo-Kantian (and Sellars inspired) view of the constitution of the necessary standards (such as mathematical rules, social practices, and/or laws of nature) that make possible objective empirical judgments.

Haugeland's view of constitution, which he further elaborates in his paper Truth and Rule Following, is intermediate between the idea of (contingent) invention of rules and discovery (of 'intelligible' or 'independent' laws).

We become aware of our own forms of life because we can compare ourselves to people in other cultures or other eras. Taking that idea deeper isn't uncontroversial, though, is it? Aren't we just speculating that there could be sentient beings who see a radically different world from our own? — frank

Much hangs on what "radically different" means. It could mean that two existing forms of life are incommensurate in such a way that mutual understanding is impossible in principle. Or a form of life could be radically different than our own in the mundane sense that it is difficult for us to fathom prior to having gained some acquaintance with it. Michael Thompson has argued that there is a multiplicity of practical forms of life, while Sebastian Rödl has argued that there is only one. Donald Davidson, in his paper On the Very Idea of a Conceptual Scheme, also argued that the very idea of a multiplicity of mutually incommensurable practical forms of life (or conceptual schemes) is incoherent. That's an issue that Rovelli doesn't contend with. It points to the possibility of a middle term between viewing mathematical theories in a way that make them inherit the contingency of the forms of life which they speak to, on the one hand, and viewing them to be universal in a way that makes them independent of (or unsoiled by) any embodied and situated life form whatsoever, on the other hand.

Hi! Would you mind presenting a quick explanation of the argument? I'll pay you in hamburgers. — frank

It's difficult for me to improve much on @fdrake's summary of Rovelli's argument, earlier in this thread. This is a broadly negative argument, however, that consists in highlighting that the version of mathematical Platonism which Rovelli is targeting incorporates too many items into the set of what Platonists themselves intuitively feel are entitled to be counted as intelligible mathematical patterns. The argument relies on the acknowledgement that the manner in which we sort out the wheat (fruitful mathematical theories) from the chaff (unprincipled and uninteresting sets of axioms) reflects contingent features of our specific form of life. This consideration, supporting the negative argument, fails however, it seems to me, to properly account for the fact that mathematical truths appear to have a grade of necessity (and degree of generality) somehow intermediate between, and qualitatively distinct from both, logical necessity and pure contingency. (I've suggested that Kant, and neo-Kantians such as Sellars, are gesturing towards the right kind of necessity with the concept of synthetic a priori propositons). But that's not what you're asking about. Maybe I'll comment more about this in another post.

But then I reasoned (while simultaneously realizing that it made no sense!) that, on the one hand, there couldn't be any horizontal force on the top screw without there being a torque on the middle screw (...) — Pierre-Normand

Maybe I should clarify my reasoning a bit here. Since the frame of the sign is being held by two screws, the force of the wind on the sign will be opposed by an equal reaction force distributed on the two screws. (I am ignoring the torque around the vertical axis, which is not relevant here). But my main point is that the force being applied on either screw doesn't result in any torque (around the horizontal axis) being applied on the other screw unless the frame, working as a lever, is allowed of rotate a little bit. And this can't happen prior to one of the screws, at least, beginning to loosen up.

If that's right, then the only concrete example used to argue against the solution that suggests we should rule out the dome as an inadmissable idealisation because of the infinite curvature at the top, has failed. All that is left to argue against that solution is the second last paragraph on p21 that begins with 'It does not.' But I found that para rather a vague word salad and didn't feel that it contained any strong points. Indeed I'm not sure I understood what point he was trying to make in it. Perhaps somebody could help me with that. — andrewk

I had also highlighted this paragraph from Norton's paper in orange, which is the color that I use to single out arguments that appear incorrect to me. In the margin, I had written "idem", referring back to my comment about the previous paragraph. On the margin of the previous paragraph, I had commented: "That may be because the limit wasn't approached correctly. A more revealing way to approach the limit would be to hold constant the shape of the dome (with infinite curvature at the apex, or close to it) and send balls sliding up with a small error spread around the apex. The limit would be taken where the error spread is being reduced."

Alternatively, we can proceed in the way @SophistiCat discussed, and allow the dome to have some elasticity. There will be an area near the apex where the mass is allowed to sink in and remains stuck. When we approach the limit of perfect rigidity, the sensitivity to the initial placement of the mass in the vicinity of the apex increases without limit and we get to the bifurcation point in the phase space representation of this ideal system. The evolution is indeterminate because, through going to the ideal limit, we have hidden some feature of the dynamics and turned the problem into a problem of pure statics (comparable to the illegitimate idealization of the beam pseudo-problem, which was aptly analogized by SophistiCat as the fallacious attempt to determine the true value of 0/0 with no concern for the specific way in which this ideal limit is being approached).

@SophistiCat, @andrewk,

I was quite startled when I read the case of the beam, described by Norton as a "statically indeterminate structure", because just a few days earlier I had been struck by a similar real world case. I was startled yet again today when I say you two insightfully discussing it.

A friend of mine went to Cuba with his wife and they asked me to feed their two cats while they were away. They live on my home street, a mere five minutes walk away. As I was walking there, I noticed the street sign, with the name of our street on it, being rotated 45° from its normal horizontal position. The sign is screwed inside of an iron frame, and the frame was secured to the wooden pole in its middle position, while another screw, at the top, had come loose and thus enabled the sign to rotate around its (slightly off center) horizontal axis. (I guess I'll go back there during the day and take a picture). A few weeks ago there had been a storm in our area, with abnormally heavy winds, which knocked the power out for several hours. Those heavy winds may have been the cause.

But then, as I wondered what kind of force might have cause the top screw to come loose, I also wondered how any torque might have been applied on the middle screw prior to the top screw giving way (or, at least, beginning to loosen up). I assumed the middle screw not to have been centered on the horizontal axis of the sign, or else there would have been no torque from the wind. But then I reasoned (while simultaneously realizing that it made no sense!) that, on the one hand, there couldn't be any horizontal force on the top screw without there being a torque on the middle screw, but also, on the second hand, that there could not be any torque on the middle screw without there being an initial horizontal displacement of at least one of the two screws! So, on the condition that the whole system would be perfectly rigid, and treated as a problem of pure statics, there appears to be no possibility for either a torque being applied on the middle screw, or an equal and opposite lever reaction force being applied on the top screw, without one of those two forces being enabled to produce (dynamically, not merely statically) an initial action, or reaction, just as SophistiCat described regarding Norton's horizontal beam case. And such an initial dynamical action only is possible on the condition that the system not be ideally rigid. And then, of course, as SophistiCat astutely concluded (and I didn't concluded at the time) the ideal case might be inderminate because of the different ways in which the limiting case of a perfectly rigid body could be approached. (I had later arrived at a similar conclusion, regarding Norton's dome, while Norton himself, apparently, didn't. But I'll comment on this shortly).

So I wouldn't worry too much about these singular limits; just as in the case of the division by zero, no solution makes more sense than any other, they are all meaningless. — SophistiCat

You mean, of course, the division of zero by zero. (Very good recent exchange between you and andrewk, by the way! I'll comment shortly.)

(...) Muddy the waters elsewhere you intellectual cretin. — StreetlightX

This is over the top and uncalled for. From where I stand, Wayfarer appears to have made some good and relevant points (which I was planning to comment on). But even is I'm wrong about that, your response still is uncalled for. Moderator, moderate yourself!

The relation between such an unconstrained world of math and a limited finite world is that the limited finite world is a part of the unconstrained world of math. — litewave

... Just like the material of Michelangelo's David was a smaller part of the whole block of marble, which it was carved out from. Would you say that, therefore, the statue already existed as a distinctive part of the whole block independently of Michelangelo's act of carving it out?

What I understand is that modern-day Platonism is more like Pythagorean idealism. Although the refutation of Pythagorean idealism is commonly attributed to Aristotle, it has been argued that Plato actually laid the grounds for this. Plato worked to expose and clarify all the principles of Pythagorean idealism, and in the process uncovered its failings. I've seen it argued that the Parmenides, though it is quite difficult to understand, serves to refute this form of idealism. — Metaphysician Undercover

Thanks very much for those reminders. That's indeed an important thread of the history of ideas to be reminded of. Maybe Aristotle shares some part of the blame for having appropriated (while he doubtlessly improved on some of them) some features of Plato's criticism of idealism while ascribing to Plato himself theses that Plato (or Plato's Socrates) was merely expounding rhetorically. Aristotle's third man argument, if I remember, which is directed by him against Plato, is actually borrowed from Plato, if I remember.

In defense of a dogma seems like a really fun article. I just started it and I'm impressed by the style, precision and generosity of the argument! — fdrake

I'm glad you like it. I think it has some relevance to the present topic since what is at issue, in Rovelli's polemics against the Platonic thesis that the domain of interesting mathematical objects might be identified with an ideal universe 'M' allegedly knowable a priori, is the contingency of our constitutive relation to what it is that we justifiably find interesting in mathematics (in such a way that it can so much as count as genuinely mathematical). Strawson and Grice, however, began loosening up, in their response to Quine, the false dichotomy between a prioricity (wrongly construed as a purely epistemic notion) and contingency (wrongly construed as a purely metaphysical notion), long before Kripke. They thereby recover some insight from Kant about the requirement for synthetic a priori propositions. (Much confusion arises, though, from inconsistencies in the use of the analytic/synthetic, a priori/a posteriory, and necessary/contingent pairs of predicates). This false dichotomy is something I'll have a bit more to say about in another post.

Suppose I infuse a needle-like intrusion to break the water's surface tension to prevent its meniscus from settling on that level of the groove and to direct water out of the tube as well? (Bear with me, I'm trying to see if I can cook a solution to these possible limitations.) — BrianW

I've edited my response and it may answer your question.

Suppose I infuse a needle-like intrusion to break the water's surface tension to prevent its meniscus from settling on that level of the groove and to direct water out of the tube as well? (Bear with me, I'm trying to see if I can cook a solution to these possible limitations.) — BrianW

Rather than using a needle, you could also lay a bit of paper towel across the top of the edge of the tube. That would break the surface of the concave meniscus as well. That would achieve the same result as a self priming siphon directing the water out of the tube.

So, maybe what you're picturing, now, is some sort of a self priming siphon continuously emptying up the top of the tube.

If you have two buckets of water at different heights, you could dip the end of a thin dry siphon into the higher bucket until it has spontaneously filled up (by mere capillary action) and starts dripping into the lower bucket. At that point, you could even sink the lower end of the siphon onto the lower bucket and the water in the higher bucket will keep flowing though the now completely wet siphon into the lower bucket until the water levels have equalized, right? But after the initial priming has occurred, the siphon never is going to carry the water from a lower level to a higher level. Furthermore, it will not self prime for carrying water from a lower to a higher level.

So, likewise with your meniscus breaking machine. You are effectively siphoning water though the meniscus, and out of the tube. But it's a self priming siphon that will stop working (or fail to start working) whenever (or if) the outside mouth of this siphon opens up at a level that is higher than the water level at the bottom of the tube. The tube and the siphon dipped in it, with the help of the surface tension of the meniscus, will effectively join the tube and the small siphon into one sigle siphon unable to carry water to a higher level.

Now you roll the old cylinder out of the way and stand up a new one, which doesn't weigh anything to speak of yet...

It would still violate conservation-of-energy, and therefore it would still be impossible.

But now it isn't quite as obvious why it wouldn't work. — Michael Ossipoff

Of course, the work that is being produced by the falling tube is extracted from the gravitational potential energy of the water that rose into the tube, and this gravitational potential energy had been produced by converting the electromagnetic potential energy from the attractive force between the water molecules and the internal surface of the tube.

You've lost me a little. I'm not implying the use of plants or plant material, it doesn't have to be cellulose or organic. I mean to imitate the capillary action in plants by constructing industrial grade (metallic or some high strength synthetic fibre) capillarity tubes. — BrianW

The point that @Michael Ossipoff was making is that capillary action is a result of molecular attraction between water and the dry surface of a tube and, hence, the work that it performs while water rises against gravity into a tube is equal to the work that is requited to dry up the tube. That's because, in order to dry up the tube, you have to work against the molecular attraction force that has made the water rise into the tube (and thus increased its gravitational potential energy) in the first place.

You may be picturing a continuous loop process in which the water circulates in the tube, such that you never need to dry the tube up. But in that case, after the tube has been initially filled up, there is no more capillary action. You would need a pump in order to maintain the upward circulation of water against gravity, and this pump would consume as much energy (at least) as you thereafter extract from the water flowing down around the turbines of your hydroelectric power plant. What plays the role of this powerful pump, in the case of living plants, is the solar radiation drying up the upper parts of the tubes within the plant leaves.

Lloyd Gerson, What is Platonism? — Wayfarer

It's funny that you would mention Gerson's book, since I added it to the Platonism folder in my digital library a few hours ago, having found it thanks to the title of the first chapter: "Was Plato a Platonist?" (which was also the title of the paper by Konrad Rokstad that I referenced earlier).

This makes sense. I don't have the knowledge to bring out how Plato became distorted, though. What history are you tracing in this idea? — fdrake

I was alerted to the possibility of the distortion by a handful of scattered remarks on Plato versus Platonism by John McDowell. But I haven't pondered much on the historical roots of the distortion, nor do I feel equipped for tracing such roots anywhere earlier than the modern period.

I just did some literature search and found this paper, which makes similar remarks to McDowell's: Konrad Rokstad, Was Plato a Platonist? Analecta Husserlania. The Yearbook of Phenomenological Research, Volume CX, 2011.

Regarding what might be seen as a modern recovery of Plato's valuable insight about the dialectical nature of the constraining function of a priori 'forms' upon reason, I made this recent comment.

It's like saying: wow, look at all these various languages that have nouns! Guess Nouns must be Platonic Entities. It's reasoning made for idiots. — StreetlightX

There is an undeniable phenomenon of convergence. But there are two mistakes one could make regarding such phenomena.

The mistake that animates modern naive empiricism is to explain the phenomenon of convergence -- such as the discovery of laws of nature, or of general logico-grammatical features shared by (most) natural languages -- as a result of the faithful (or approximate) reproduction, as contents of our mental representations, of the structure of an independently existing empirical ('external') reality. This mistake is almost indistinguishable from the mistake of reifying intelligible ideas, where such ideas are being ascribed a role similar to the role being played by raw 'sense data' for the empiricist.

Another mistake would be to recognize the root of the convergence as a product of shared social practices but to view the fact of this convergence as a matter entirely of socially enforced conventions.

One might avoid both mistakes though recognizing that the phenomenon of convergence is a dynamical product of the enactment of the social practice of arguing for or against doing and/or believing things. The convergent phenomena are being constituted from within the enactment of dialectical reason, by social beings, rather than from without their historically situated lots of shared discursive practices.

We don't just not care about them for reasons of utility, we don't care about them because we have a standard of intelligibility which automatically excludes them from our mathematical discourse. — fdrake

I think that's a point Plato would readily have acknowledged; and a reason why Plato may never have been a Platonist in the modern sense of the term. Modern mathematical Platonism likely is a distortion of Plato's thought, which distortion arose from taking his metaphors literally and misconstruing acts of the intellect -- themselves always portrayed by Plato as outcomes of strenuous and protracted dialectical effort -- as passive acts of contemplation of an independently constituted domain.

So perhaps 'mathematical world, M', is really just a metaphorical depiction of the Platonist intuition of the nature of numbers. But then, it is 'the existence of M' that is thrown into doubt. But maybe this doesn't do anything more than show that this particular way of allegorising Platonism is what is at fault. — Wayfarer

That's my suspicion too!

But anyway, the thrust of the argument is: if we took the results of all possible axiomatic systems, agglomerated them into one giant object, then granted that object independent existence - what would it look like? It would contain all kinds of bizarre crap, navigating through this world you'd hardly ever find an axiomatic system which resembled anything like our own. — fdrake

Yes, so far I have focused on a different part of Rovelli's argument that seems decisive to me but may be less intuitively compelling than his main point, which you are now drawing back the focus on. Thank's for highlighting it.

As long as there are any objects in the external reality, there are also relations between them, in the external reality. Relations and the objects between which they hold are inseparable. — litewave

Talk of the 'external reality' is quite Cartesian sounding. Cartesian materialism may be some sort of a variety of Platonism. But the relationship between representationalism in philosophical accounts of mental content, reference and truth, on the one hand, and Platonism regarding universals, on the other hand, is rather complex. I'll come back to this conversation tonight.

Relations are objects that hold between other objects (those other objects may be relations or non-relations). Relations are inseparable from the objects between which they hold. — litewave

This appear close to Russell's theory of concepts, relations and (Russellian) propositions. It is a quite Platonic theory, so, relying on it would also beg the question.

My point was there are reasons to think the structures and relations we use math to model exist in the world independent of us, since they led to us existing. — Marchesk

Interestingly enough, the fact that they enabled us to exist already begins to establish a conceptual dependence between them and us. This is not to deny that they aren't causally dependent on us. They indeed aren't. But merely to establish the causal independence that past material instantiations of formal structures (such as subsumption of past events under laws) have from us falls short from securing their conceptual autonomy in Platonic heaven.

Why would it only be a certain way for us? Do we really think that evolution or general relativity is a certain way for us, as opposed to being a certain way for the universe? — Marchesk

It's a certain way for us because we are co-evolved with our environment, or Umwelt (as Uexküll uses the term in the context of ethology, but as it can be extended to the context cultural evolution as well). Evolution isn't any way for the universe because the universe is quite dumb and doesn't care about anything in particular, not even its own material unfolding.

Arguably, we reason the way we do because the world is certain way for us to reason about it. — Marchesk

That's true, but the recognition that the world is a certain way for us to reason about it already amounts to acknowledging a productive role for the cognizing subject in constituting it and hence is a move away from Platonism.

The most general definition of mathematics I know is that it is a study of structures/relations. — litewave

Sure, but what sorts of things are structures and relations? Do they exist in themselves rather like intelligible forms in Platonic heaven? If you assume that they are universals that exist by themselves, quite independently from the constitutive roles of our practices of reasoning and discussing about them, then, in that case, you are begging the question in favor of mathematical Platonism.

@StreetlightX Although I hadn't found the time to comment, I had very much enjoyed the OP. I just now finished reading a paper by Sebastjan Vörös and Michel Bitbol -- Enacting Enaction: A Dialectic Between Knowing and Being -- recently published in the journal Constructivist Foundation. (I love the format of this journal which, like Behavioral and Brain Science, publishes target articles followed by several peer commentaries and then a response by the original author(s))

In the authors' response to the commentaries, they produced a quote from Varela that reminded me of your OP.

'As Varela himself pointed out, “ideas appear as movements of historical
networks in which individuals are formed, rather than vice versa,” and tracing a genesis of a given idea is like “making a fold in history where men and ideas live because we are points of accumulations among the social networks in which we live” (Varela 1996b: 408; our emphasis).'

Breifly, on this - Do you know if this is something that is in Daniel Everett's discussion of the Piraha language? — StreetlightX

I have Everett's book in my digital library but I am yet to read it too. The case of the Piraha and their (lack of) counting abilities had been a topic of discussion very many years ago on the now defunct (but still archived) Yahoo 'analytic' discussion group. From what I remember, there is an interpretive disagreement between the psychologist Peter Gordon, who also studied the Piraha (lack of) numerical abilities and Everett, who is more of a Chomskyan linguist and, hence, who is less inclined to take seriously the Whorf-Sapir hypothesis on the essential link between language and cognition. I'm siding more with Gordon's analysis than with Everett's, because of this issue, regarding this specific topic. (So, you might also be interested in digging up Gordon's relevant publications)

Let me add that, here on Earth, we have the Pirahas of the Amazon rainforest who don't have any use for natural numbers, not even the number 1, nor of the existential or universal quantifiers (which Hume, I think, argued were required for grounding the practice of counting).

Rovelli's argument is compelling to me. His view on natural numbers, in particular, meshes rather well with Frege's construal of them as second order functors that are predicated of first order functors (e.g. sortal concepts) where those first order concepts do the prior work of individuating the objects to be counted. If Jupiterians don't have a need for the sorts of first order functors that we make use of to individuate discrete persisting objects, then they wouldn't have much use for our concept of a natural number either. They maybe would have a use for a somewhat isomorphic concept, however, for purposes others than counting discrete persisting material entities. Some language games are structurally (almost) identical to other language games that have very different uses or pragmatic points.

At this stage, a Platonist mathematician might insist that the concept of a natural number latches on the structural invariant shared between two such language games. But I don't think such a defense would work since the variations in the pragmatic points of structurally similar language games being played with symbolic numerals would lead to variations in the way they are structured 'around the edge' as it were. They might not be axiomatized quite in the same way nor be projectable in the same manner to extended domains. (I'd have to conjure up some example that might be more compelling than Kripke's 'quus' example).

Mm, I'm of a know-thy-enemy type as well. You fight cancer by studying it rigorously and prodding it incessantly. The Sophist remains one of my favorite philosophical works. — StreetlightX

Great!

And the Plato I have in mind is more the Plato who valorizes eternity, who rejects becoming, and poses infantile questions. — StreetlightX

Fair enough. I have a tendency, myself, to try to pay attention to the best in the thought and hidden legacy of influential philosophers. This includes Descartes, whose influence I oppose relentlessly, but whose better insights have been aptly advocated by Daniel Robinson (who is nevertheless closely philosophically allied with the arch-Wittgensteinian and arch-anti-Cartesian P.M.S. Hacker!)

I'm not familiar with Goodman's new problem — csalisbury

It's not very new, though. The New Riddle of Induction is the fourth chapter in Goodman's book Fact, Fiction and Forecast, first published in 1955 and adapted from lectures given in 1953. It is related to Kripke's 'quus' alternative rule for addition discussed in his Wittgenstein on Rules and Private Language (1982).

Platonism is philosophical cancer. — StreetlightX

On the other hand, you are saluting the nominalistic proclivities of Sellars, Quine and Davidson, among others. Let me just focus on Sellars for now.

Sellars's nominalism (if we can call it that) operates through Kant, and pragmatizes it. It is especially indebted to Kant for its acknowledgement of the constitutive and ineliminable role of our conceptual powers in shaping up experience (while recognizing the equally indispensable reciprocal role of sensory intuitions, or receptivity, in the constitution of the power of judgement). It is epitomized, within Sellars' own neo-Kantian pragmatism, by its denunciation of the Myth of the (non-conceptual) Given. But it also jettisons the idea that the a priori forms of the understanding can be disclosed by means of pure armchair exercises of the power of the intellect.

So, Sellars's pragmatism is inimical to Plato, in that sense. But it still retains, from Plato (and through Kant), the idea that intellectual reflection can reveal to the intellect its own a priori forms. Those a priori forms, however, are conceived within Sellars's pragmatized neo-Kantianism rather more on the model of Wittgenstein's grammatical remarks, or hinge propositions, or the constitutive laws/rules of embodied, situated and historicized scientific practices, or in the way Strawson and Grice have conceived of "analytic" statements (comparable to Sellars's synthetic a priori statements) in their rejoinder to Quine's 1951 Two Dogmas of Empiricism (in In Defense of a Dogma, 1956).

I believe that an "event" is completely artificial, in the sense that "an event" only exists according to how it is individuated by the mind which individuates it. So the problem you refer to here is a function of this artificiality of any referred to event. It is a matter of removing something form its context, as if it could be an individual thing without being part of a larger whole. — Metaphysician Undercover

I agree with this. I am indeed stressing the fact that the event doesn't exist -- or can't be thought of, or referred to, as the sort of event that it is -- apart from its relational properties. And paramount among those constitutive relational properties are some of the intentional features of the 'mind' who is individuating the event, in accordance with her practical and/or theoretical interests, and embodied capacities.

(...) And when we see things this way we have to ask are any events really accidental or coincidental. it might just be a function of how they are individuated and removed from context, that makes them appear this way.

Yes, indeed. The context may be lost owing to a tendency to attempt reducing it to a description of its material constituent processes that abstracts away from the relevant relational and functional properties of the event (including constitutive relations to the interests and powers of the inquirer). But the very same reductionist tendency can lead one to assume that whenever a 'composite' event appears to be a mere accident there ought to be an underlying cause of its occurrence expressible in terms of the sufficient causal conditions of the constituent material processes purportedly making up this 'event'. Such causes may be wholly irrelevant to the explanation of the occurrence of the composite 'event', suitable described as the purported "meeting" of two human beings at a well, for instance.

There is a clear problem with this example, and this is the result of expecting that an event has only one cause. When we allow that events have multiple causes, then each of the two friends have reasons (cause) to be where they are, and these are the causes of their chance meeting. — Metaphysician Undercover

It's true, in a sense, that 'events' have multiple causes. Recent work on the contrastive characters of causation and of explanation highlight this. But what it highlights, and what Aristotelian considerations also highlight, is somewhat obscured by the tendency in modern philosophy to individuate 'events' (and hence, also, effects) in an atomic manner as if they were local occurrences in space and in time that are what they are independently from their causes, or from the character of the agents, and of their powers, that cause them to occur. This modern tendency is encouraged by broadly Humean considerations on causation, and the metaphysical realism of modern reductionist science, and of scientific materialism.

If we don't endorse metaphysical realism, then we must acknowledge that the event consisting in the two acquaintances meeting at the well can't be identified merely with 'what happens there and then' quite appart from our interest in the non-accidental features of this event that we have specifically picked up as at topic of inquiry. Hence, the event consisting in the two individuals' meeting can't be exhaustively decomposed into two separate component events each one consisting in the arrival of one individual at the well at that specific time. The obvious trouble with this attempted decomposition is that a complete causal explanation of each one of the 'component events' might do nothing to explain the non-accidental nature of the meeting, in the case where this meeting indeed wouldn't be accidental. In the case where it is, then, one might acknowledge, following Aristotle, that the 'event' is purely an accident and doesn't have a cause under that description (that is, viewed as a meeting).

So the event, the chance meeting, is caused, but it has multiple causes which must all come together.

Well, either it's a chance encounter or it's a non-accidental meeting. Only in the later case might a cause be found that is constitutive of the event being a meeting (maybe willed by a third individual, or probabilistically caused by non-accidental features of the surrounding topography, etc.)

When we look for "the cause", in the sense of a single cause, for an event which was caused by multiple factors, we may well conclude that the event has no cause, because there is no such thing as "the cause" of the event, there is a multitude of necessary factors, causes.

Agreed. Those separate causes, though, may explain separately the different features of the so called 'event' without amounting to an explanation why the whole 'event', as such, came together, and hence fail to constitute a cause for it (let alone the cause).

I think we meant different things by indeterminism. In the paper's sense of 'a single past can be followed by many futures', the translational time symmetry of the non-zero solution is what facilitates that conclusion. If the ball decides to fall in a given direction, its behaviour is determined at every point on that path by the equations of motion (after redefining t-T=0). — fdrake

Yes, because the path of the system in phase space only is branching out at the point representing the particle being at rest at the apex. When the particle is already has acquired some momentum, some distance away from the apex, then its trajectory is fully determined in both time direction up to the point where it gets to (or came from) the bifurcation point (that is, to the apex, at rest).

By 'determinism', as predicated of a material system and its laws, I only mean that this system's state at a time (and the laws) uniquely determines its state at any other time. That a single past (either a single past instantaneous state, or a single past historical trajectory in phase space) leads to a unique future is a corollary.

So your claim that "the environment" is an acting agent, is nonsense without some principles whereby "the environment" can be conceived as an acting, unified whole. — Metaphysician Undercover

Apocrisis was talking about a generic force rather than a generic cause, or generic agent. I think is makes sense to speak of a general background condition that isn't happily conceived of as a cause of the events that they enable to occur (randomly, at some frequency). Causes ought to be explanatory. So, there may be events that are purely accidental and, hence, don't have a cause at all although they may be expected to arise with some definite probabilistic frequencies. Radioactive decay may be such an example. Consider also Aristotle's discussion of two friends accidentally meeting at a well. Even though each friend was caused to get there at that time (because she wanted to get water at that time, say), there need not be any cause for them to have both been there at the same time. Their meeting is a purely uncaused accident, although some background condition, such as there being only one well in the neighborhood, may have made it more likely.