OK, I think I understand. The way you tell it, you seem to think that bodies are driven entirely by their internal processes. And yet you have also mentioned external forces - doesn't that begin to answer your question?

Let's imagine a setup with a potential field and a particle that interacts with that field (e.g. think of a rock falling from a height). Assume that the particle has no parts; it still responds to that field and does something (maybe not anything very interesting, but hey, it's just one particle). Does this solve the problem?

Well, what I am trying to understand is why you are framing the problem in mereological terms. You say that "without component parts, it is hard to imagine how..." So would it be easier to imagine with component parts?

I am not even doing the Socratic thing here, I am genuinely not understanding your thought process.

I am not sure why you think that "irreducible particles" in particular present a difficulty with respect to interaction and complex behavior. How do you think compound bodies manage it? What is it about irreducible particles that should prevent them from interacting and behaving?

If you look at my article, "Mind or Randomness in Evolution" (https://www.academia.edu/27797943/Mind_or_Randomness_in_Evolution) — Dfpolis

Thank you. Having looked over your article, I have no further interest in this conversation.

The way a Wager-like argument is commonly used (again, it is arguable whether Pascal himself meant it that way), the argument seeks to sidestep the burden of justifying its ontological claim; the cost/benefit analysis, as stated in the argument, is supposed to be doing all the heavy lifting. That is the main selling point of the argument. But we have concluded, at least for your version, that the burden of providing a convincing argument for the existence of God cannot be avoided.

1.The beauty of the universe is improbable under atheism
2.The beauty of the universe is not improbable under theism.
3.If we have two hypotheses and some evidence is not improbable under the first hypothesis but is improbable under the second, then that evidence counts as evidence for the first hypothesis.
4.Therefore, the beauty in the universe counts as evidence for God’s existence (1,2,3 Modus Ponens) — Empedocles

Others have developed an objection along the lines that our sense of beauty is not independent of the universe's constitution, so it is no coincidence that we see beauty in the universe.

In addition, there are more general considerations that also apply to the original fine-tuning argument, which have to do with the nature of the inference - your premise 3.

Atheism places no constraints on the physical universe - in other words, atheism is completely uninformative with regards to its constitution. Therefore, under the assumption of atheism, the probability of any given feature of the universe is either inscrutable or infinitesimal (depending on who you talk to). This goes for the laws of nature, fundamental constants, initial conditions, and anything else, including beauty. But the flip side of this is that evidence of the universe's constitution is uninformative with respect to atheism. Evidence can only count for or against a hypothesis when there is some structural relationship between the two, but not otherwise.

How about the God hypothesis - is there any relationship between it and beauty? Well, I suppose there is, but that's just because you say so. Your God hypothesis is fashioned in such a way as to make beauty likely - it is ad hoc in this way. Ad hoc hypotheses have the annoying property that they automatically raise the likelihood of whatever evidence they were fine-tuned for. For example, looking out of the window and seeing a car with a particular license number passing along the road is a very unlikely event, assuming that there is no particular reason for a car with that number being there at that moment. But if, for example, you hypothesize that God intended you to see that number at that particular moment, then the event is pretty much inevitable! Does this mean that your observation of the license plate favors God (and disfavors atheism)? Of course not. Ad hoc hypotheses, depending on who you talk to, are either inadmissible, or else they pay a high penalty in credibility that cancels out any boost they can get from evidence.

I was telling what happened, not boasting. — Dfpolis

If you say so :roll:

Any system that exhibits any regularity has "telos" in this sense, but so what? Any connection to intelligence is far from obvious. — SophistiCat

I am glad that we agree. But, if biological systems do tend toward determinant ends — Dfpolis

As do all systems (the concept of a system already implies some degree of orderliness). If telos characterizes everything in existence, simply in virtue of the definition that you give it, then it is a vacuous concept. Your analysis of teleology is wholly inadequate, or rather it is wholly absent. Once again, I recommend that you actually read something on the subject - you may learn something interesting, even if you don't agree with all of it. It's much more fun than railing against imaginary "naturalists," at least as far as I am concerned.

The point in question was special pleading by naturalists on the principle of sufficient reason. My position, stated by Freud in The Psychopathology of Everyday Life, is that if you allow any exception to the principle, you undermine the whole structure of science. — Dfpolis

It occurred after no one could rebut my argument for the existence of God in a manner consistent with the foundations of science. — Dfpolis

That's just as I said: your ideas about science and the PSR are idiosyncratic, and I expect that you will find few allies, regardless of their position on naturalism. And when you add boasts like this, you, frankly, sound like a crank. If you want to make a persuasive case, you don't want to explicitly hinge it on extreme foundational positions that few are likely to accept as an unconditional ultimatum.

It depends on what you mean by "supernatural and theological explanations." — Dfpolis

I mean the kind of explanations that hinge on the existence of a powerful and largely inscrutable personal agent.

It is clear from physics, chemistry and biology that many systems have a potential to a determinate end. That is all it means to have a telos — Dfpolis

I strongly suspect it is because they see telloi as strong evidence of intelligence — Dfpolis

That is far too general to be of much use. Any system that exhibits any regularity has "telos" in this sense, but so what? Any connection to intelligence is far from obvious.

I think you're right. Maybe if I qualified premise 1 to say something like, "If the stakes of a belief are high and credible, then you should take arguments regarding that belief seriously" then it might work? I think it's pretty intuitive that stakes play an important role in how highly we prioritize something (e.g. I am more nervous for a piano recital than a practice session, I run faster if I'm being chased by a bear, I work harder when my boss is around, etc...), so I'm hesitant to throw that idea out. — Empedocles

Yes, making the threat credible would help, but that means that the claim no longer justifies itself, which is what constitutes the principal appeal of popular invocations of Pascal's Wager; you still have to all the usual epistemological work of justifying the credibility of your claim.

At most, high stakes can serve as a lever, a force multiplier: If you present a convincing case for your claim, then I would be obliged to take it seriously and act on it. Which is what the original Pascal's Wager seeks to do, in my opinion.

I am sorry for offending you. — Dfpolis

That's ok, I wasn't really offended, and looking back, my sharp tone was unnecessary.

It was based on my experience of discussions with naturalists. Some have even rejected the foundations of science in order to maintain their faith positions. — Dfpolis

Given that your ideas of what constitutes foundations of science are rather idiosyncratic, I suspect that what you interpret as patent irrationality in the service of "maintaining faith positions" is simply a case of disagreement over those matters.

Anyway, I am not surprised at the hostile reception from self-professed naturalists who engage with you in Youtube comments. Teleology, rightly or wrongly, is commonly associated with intelligent agency, making it a poor fit for anything that doesn't have to do with human psychology, except in the context of supernatural and theological explanations. I myself am not entirely sold, not so much on teleology, as on the importance of the controversy. Some of it is merely semantic. And some, like teleology vs. teleonomy, just doesn't present high enough stakes in my mind. But that is probably because I view the issue in the epistemological plane, more than in the ontological one, and in epistemology I favor pluralism. So it's no big deal for me to accept teleological-sounding modes of explanation where they make sense.

Have I made some specific error of biological fact, or ignored some obvious rejoinder? If so, I welcome your correction. — Dfpolis

Your mistake is to charge in like a culture warrior, thinking that naturalists would necessarily be on the opposite side of the barricades, whereas much of the conversation about teleology appears to be secular. Why would a naturalist have an issue with a complex systems analysis of teleology, for example?

By not engaging, you confirm me in my position that we are discussing a faith position, not a rational conclusion. — Dfpolis

My point was that if you want to engage those whom you want to convince, you don't want to open the discussion by poisoning the well with such an obnoxious and unfair accusation. Unfair because, your famous biologist brother notwithstanding, you don't appear to be familiar with secular thought on this subject.

Your response is... to double down on the obnoxiousness. So yes, I would rather engage with Ruth Millican, or Mark Bedau, or Pierre-Normand for that matter.

What I am suggesting is that the selection process is teleological in the very same sense in which the organism's physiological and behavioral activities are teleological (or structured by means/end relationships), and for the very same reason. An organism, for instance, engages in some sort of behavior in order to quench its thirst. If it tends to succeed, thanks to some heritable feature of its physiology or anatomy, then this feature tends to be positively selected. And the reason why descendants thereafter exhibit this feature, and have the ability to engage in the behavior that such structures enables, is precisely because they subserve the end that was being actively pursued by the ancestor: namely, quenching its thirst. I conclude that the process of evolution through natural selection does have a telos, but that telos isn't external to the life form of the evolving organism; it is rather internal to it. The main engine of evolution is the organism's already existing struggle to flourish and survive (in very specific ways) in its day to day existence. — Pierre-Normand

I think you get biological teleology wrong. The way you describe it, teleology arises from individual organisms' striving to achieve a goal, much like Lamarck thought that when a giraffe reaches for higher branches its neck grows progressively longer with each generation (he even hypothesized a causal mechanism for this: a "nervous fluid"). We know that this is not how evolution works (for the most part). Fitness does not increase as a direct response to organisms' strivings and desires. As you say, the internal teleology of evolution is in the process itself, not in some agent's mental attitude. The process of evolution by natural selection molds successive populations to better fit their environment, and that works just as well on a moth's passive mimicry as on a giraffe's active feeding. It works even on unconscious, inanimate things that are similarly subject to natural selection!

I think the key to the controversy over biological teleology is in what Bedau called "mentalism:" psychologizing teleology, thinking of teleology only as a pattern of thought. Proponents of the concept of biological teleology argue that there is a more general pattern that manifests itself in other, non-psychological domains, and that both evolutionary adaptation and organismal functions are valid examples of this pattern.

Meanwhile, biology students are taught to eschew talk of biological ends. — Dfpolis

I don't know about that. I was never a biology student (and neither were you, AFAIK), so I don't know what students are taught; but teleology in biology is a controversial but well-explored topic. I wonder whether you are actually familiar with any of that discussion.

Philosophical naturalists reject finality, not because doing so is rational, but because it threatens their faith position — Dfpolis

Oh boy. You know, when you write something as obnoxious as that, one is discouraged from reading further.

In the context of this topic (purported normativity of logic), "logic" is not just any abstract system of inference. You could throw together any number of such systems; obviously, they couldn't all be normative. (Normative for what?) The "logic" under consideration is what underlies ordinary deduction. Sometimes people talk of the FOPL in this context, but I think that FOPL can only serve as a proxy. What is (purportedly) normative is an unformalized intuitive rationality, which can be brought out with simple, uncontroversial examples of deduction. Formal logic, such as the modern-day FOPL, aims to capture this intuitive rationality.

Logic does not just set out how we ought speak, but how we can speak. It shows us what sorts of speaking are wrong. — Banno

You think you are saying something different, but you are not. You just understand "normative" in a narrow ethical sense.

Good question. I have never had a clear idea of what people mean by normative, and looking up definitions doesn't seem to help. — andrewk

I would define a normative belief as one that directly justifies and urges or inhibits action or conduct. It says "You should do this," "This is the right way" or "That is the wrong way." A normative principle is a generalization of such beliefs. When I say "directly" I mean that the normative belief does not acquire its normative status in virtue of something else. It doesn't tell you that you should do this or that for reasons, because if there are other reasons, then they are in turn the source of normativity. Normative beliefs are where reasons bottom out.

The condition that a normative principle applies not just to oneself, but to everyone is a contingent, non-essential property of normative principles, I think. Normative principles are often, but not always universal. Moral relativists typically believe that the force of moral principles is limited to a particular culture or community. Jews believe that their religious strictures apply only to Jews (there are a few Biblical prescriptions applicable to gentiles as well, known as Noahide laws). Personal obligations and vows have force only for one particular person.

I realize that all this is somewhat vague and squishy, and will fray and crumble at the edges when subjected to scrutiny, as analytical philosophers especially like to do. But that just means that the concept of normativity is not simple, clean and sharp-edged, which is to be expected for a concept whose nature is psychosocial.

One can raise the stakes of any proposition, simply by appending to it a stake-raising clause. For any proposition A there is a proposition A* = A & C, where C = (costly consequences for not believing A). So the consequence of your position is that you have to take seriously infinitely many propositions, not just those concerning the existence of god and his proper worship.

You could object that A* is spurious, but your argument does not say anything about the content and the quality of the propositions that you say should be taken seriously; the only reason you give for taking them seriously C.

Try to mount an argument that we ought to use logic if we wish to arrive at true beliefs, without using logic. — andrewk

Isn't that what the normative principle amounts to (or something similar)? It's precisely because mounting an argument without the use of logic is impossible by definition that such guiding principles cannot be anything other than normative.

It's mind-boggling that despite a consensus opinion of the expert community, no remotely credible scientific and engineering justification, no independently verified demonstrations, and no apparent interest from the industry, which would stand to profit enormously if LENR was viable, there are still people eager to uncritically swallow this bullshit year after year after year. "Coming to market soon!" (And we've been hearing this song for how long? From Rossi alone - since 2011 at least.)

Fascinating factoid: In a single hour, the amount of power from the sun that strikes the Earth is more than the entire world consumes in an year — Hanover

But in order to make use of all that energy you would have to cover the entire surface of the earth with perfectly efficient solar panels, which then perfectly efficiently deliver that energy to end consumers. So, all things considered, that doesn't actually seem like that much usable energy.

In The Norton Dome and the Nineteenth Century Foundations of Determinism, 2014 (PDF) Marij van Strien takes a look at how 19th century mathematicians and physicists confronted instances of indeterminism in classical mechanics*. She notes that the fact that differential equations of motion sometimes lacked unique solutions was known pretty much for as long as differential equations were studied, since the beginning of the 19th century. Notably, in 1870s Joseph Boussinesq studied a parameterized dome setup, of which what is now known as the Norton's Dome is a special case. Although Boussinesq did not solve the equations of motion for the dome, he studied their properties, and noting the emergence of singularities for some combinations of parameters he concluded that the equations must lack a unique solution in those instances (this can also be seen by noting that the apex of the dome does not satisfy the Lipschitz condition in those configurations - see above.)

* A less technical work by the same author is Vital Instability: Life and Free Will in Physics and Physiology, 1860–1880, 2015 (PDF).

Van Strien writes that "nineteenth century determinism was primarily taken to be a presupposition of theories in physics." Boussinesq was something of an exception in that he took the nonunique solutions that he and others discovered seriously. (He acknowledged that his dome was not a realistic example, taking it more as a proof-of-concept; rather, he thought that actual indeterminism would be found in some hypothetical force fields produced by interactions between atoms, which he showed to be mathematically similar to the dome equations.) Boussinesq believed that such branching solutions of mechanical equations provided a way out of Laplacian determinism, giving the opportunity for life forces and individual free will to do their own thing.

But by and large, Van Strien says, such mathematical anomalies were not taken as indications of something real: in cases where solutions to equations of motion were nonunique, one just needed to pick the physical solution and discard the unphysical ones. This is probably why these earlier discoveries did not make much of an impression at the time and have since been partly forgotten, so that Norton's paper, when it came out, caused a bit of a scandal.

From my own modest experience, such attitudes towards mathematical models still prevail, at least in traditional scientific education and practice. It is not uncommon for a model or a mathematical technique to produce an unphysical artefact, such as multiple solutions where a single solution is expected, a negative quantity where only positive quantities make sense, forces in an empty space in addition to forces in a medium, etc. Scientists and engineers mostly treat their models pragmatically, as useful tools; they don't necessarily think of them as a perfect match to the structure of the Universe. It is only when a model is regarded as fundamental that some begin to take the math really seriously - all of it, not just the pragmatically relevant parts. So that if the model turns out to be mathematically indeterministic, even in an infinitesimal and empirically inaccessible portion of its domain, this is thought to have important metaphysical implications.

Interpretations of quantum mechanics are another example of such mathematical "fundamentalism". Proponents of the Everett "many worlds" interpretation, such as cosmologist Sean Carroll, say (approvingly) that all their preferred interpretation does is "take the math seriously." Indeed, the "worlds" of the MWI are a straightforward interpretation of the branching of the quantum wavefunction. (Full disclosure: I myself am sympathetic to the MWI, to the extent to which I can understand it.)

Are "fundametalists" right? Can a really good theory give us warrant to take all of its implications seriously?

"Nuclear fusion is always 30 years away," as they say. This indeed has been the case for the last half-century if not more.

LENR is just a rebranding of cold fusion. It is largely the province of cooks and scammers.

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. — Pierre-Normand

Thanks, but I cannot take credit for what I didn't actually say :) The rigid beam case is indeterminate in the sense that multiple solutions are consistent with the given conditions. I did think that there may be a way to approach a different solution (with non-zero lateral forces) by an alternative path of idealization, perhaps by varying something other than elasticity. My intuition was primed by the 0/0 analogy, in which a parallel strategy of approaching the limit from some unproblematic starting point would clearly be fallacious. But I couldn't think of anything at the moment, so I didn't mention it.

I think I have such an example now though.



Suppose that a pair of lateral forces of equal magnitude but opposite directions were applied to the beam. This would make no difference to the original rigid beam/wall system: the forces would balance each other, thus ensuring equilibrium, and everything else would be the same, since the forces wouldn't produce any strains in the beam; the forces would thus be merely imaginary, since they wouldn't make any physical difference. However, if we were to approach the rigid limit by starting from a finite elasticity coefficient and then taking it to to infinity, there would be a finite lateral force acting on the walls all the way to the limit.

Korolev actually does a similar limiting analysis of the dome itself, showing that for any finite elasticity (assuming a perfectly elastic material of the dome) there is a unique solution for a mass at the apex: the mass remains stationary; but when the elasticity coefficient is taken to infinity, i.e. the dome becomes perfectly rigid, the situation changes catastrophically and all of a sudden there is no longer a unique solution.

In the case of the beam in Norton's paper, in order to definitely answer the question we need to find the state of stress inside the beam, which for normal bodies is fixed by boundary conditions. The problem with infinite stiffness is that constitutive equations are singular, and therefore, formally at least, the same boundary conditions are consistent with any number of stress distributions in the beam. This happens for the same reason that division by zero is undefined: any solution fits. In the simple uniaxial case, the strain is related to the stress by the equation $\epsilon = \frac{1}{E}\sigma$. When E is infinite and 1/E is zero, the strain is zero and any stress is a valid solution to the equation. Boundary conditions in an extended rigid body will fix (some of the) stresses at the boundary, but not anywhere else. Or so it would seem.

However, what meaning do stresses have in an infinitely stiff body? Because there can be no displacements, there is no action. Stresses are meaningless. Suppose that instead of perfectly rigid and stationary supports, the rigid beam in Norton's paper was suspended between elastic walls. Whereas in the original problem the entire system, including the supports, was infinitely stiff and admitted no displacements, now elastic walls would experience the action of the forces exerted by the ends of the beam. Displacements would occur, energy would be expended, work would be done. The problem becomes physical, and physics requires energy conservation, which immediately yields the solution to the problem.

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.

In Determinism: what we have learned and what we still don't know (2005) John Earman "survey[ s] the implications of the theories of modern physics for the doctrine of determinism" (see also Earman 2007 and 2008 for a more technical analysis). When it comes to Newtonian mechanics he gives several indeterministic scenarios that are permitted under the theory, including the Norton's dome of the OP. Another well-known example is that of the "space invaders":

Certain configurations of as few as 5 gravitating, non-colliding point particles can lead to one particle accelerating without bound, acquiring an infinite speed in finite time. The time-reverse of this scenario implies that a particle can just appear out of nowhere, its appearance not entailed by a preceding state of the world, thus violating determinism.

A number of such determinism-violating scenarios for Newtonian particles have been discovered, though most of them involve infinite speeds, infinitely deep gravitational wells of point masses, contrived force fields, and other physically contentious premises.

Norton's scenario is interesting in that it presents an intuitively plausible setup that does not involve the sort of singularities, infinities or supertasks that would be relatively easy to dismiss as unphysical. has already homed in on one suspect feature of the setup, which is the non-smooth, non-analytic shape of the surface and the displacement path of the ball. Alexandre Korolev in Indeterminism, Asymptotic Reasoning, and Time Irreversibility in Classical Physics (2006) identifies a weaker geometric constraint than that of smoothness or analyticity, which is Lipschitz continuity:

A function $f$ is called Lipschitz continuous if there exists a positive real constant $K$ such that, for all real $x_1$ and $x_2$, $|f(x_1) - f(x_2)| \le K |x_1 - x_2|$. A Lipschitz-continuous function is continuous, but not necessarily smooth. Intuitively, the Lipschitz condition puts a finite limit on the function's rate of change.

Korolev shows that violations of Lipschitz continuity lead to branching solutions not only in the case of the Norton's dome, but in other scenarios as well, and in the same spirit as Andrew above, he proposes that Lipschitz condition should be considered a constitutive feature of classical mechanics in order to avoid, as he puts it, "physically impossible solutions that have no serious metaphysical import."

Ironically, as Samuel Fletcher notes in What Counts as a Newtonian System? The View from Norton’s Dome (2011), Korolev's own example of non-Lipschitz velocities in fluid dynamics is instrumental to productive research in turbulence modeling, "one whose practitioners would be loathe to abandon on account of Norton’s dome."

It seems to me that Earman oversells his point when he writes that "the fault modes of determinism in classical physics are so numerous and various as to make determinism in this context seem rather surprising." I like Fletcher's philosophical analysis, whose major point is that there is no uniquely correct formulation of classical mechanics, and that different formulations are "appropriate and useful for different purposes:"

As soon as one specifies which class of mathematical models one refers to by “classical mechanics,” one can unambiguously formulate and perhaps answer the question of determinism as a precise mathematical statement. But, I emphasize, there is no a priori reason to choose a sole one among these. In practice, the choice of a particular formulation of classical mechanics will depend largely on pragmatic factors like what one is trying to do with the theory. — Fletcher

Yes indeed. They are abrogating their moral duty to the letter of the law. Which makes them less human, allegedly. — unenlightened

Not exercising a quintessentially human faculty in the context of performing a specific task does not make you "less than human," in and of itself - it all depends on context. Following printed instructions while assembling an IKEA table won't land you in Nuremberg. What is morally suspect is abrogating moral duty when exercising moral duty is called for (and yes, that can happen in a legal context as well).

Does this apply to judges who refer to statute, convention, constitution, case law, etc? — unenlightened

When judges defer to law, they are not exercising their human ethical judgment (at least in theory, which I take to be the context of your hypothetical question).

No, it's not just a semi-infinite number line, because that omits the temporal context. Time does not exist all at once, as does an abstract number line.

Consider the future: it doesn't exist. — Relativist

Neither does the past, whether finite or infinite, according to the A theory of time, which you brought up for no apparent reason. The A theory of time is a red herring; this metaphysical position is irrelevant to the argument that you are trying to make, which is:

The present is the END of a journey of all prior days. That would be the mirror image of reaching a day infinitely far into the future, which cannot happen. A temporal process cannot reach TO infinity, and neither can a temporal process reach FROM an infinity. — Relativist

We've been over this already: this is the same question-begging argument that you made at the beginning of the discussion. The reason a temporal process will never reach infinitely far into the future is that there is nothing for it to reach: a process can start at point A and reach point B, but if there is no point B, then talk about reaching something doesn't make sense. Turn this around, and you get the same thing: you can talk about reaching the present from some point in the past, but if there is no starting point (ex hypothesi), the talk about reaching from somewhere doesn't make sense, unless you implicitly assume your conclusion (that time has a starting point in the past).

Look, you don't have an argument here; you are just stating and restating your conclusion in slightly different ways. You aren't the first to fight this hopeless fight, of course: the a priori denial of actual infinities is as old as Aristotle; Kant tried to make an argument very similar to yours, and others have followed in his step, including most recently theologian W. L. Craig, who employs a raft of such arguments as part of his Kalam cosmological argument for the existence of God. But nowadays these arguments do not enjoy much support among philosophers (see for instance Popper's critique, if you can get it, or any number of more recent articles).

As for physicists and cosmologists, to whom you have appealed as well, most don't even take such a priori arguments seriously, though a few have condescended to offer a critique (such as the late great John Bell, back in 1979, responding to the same article as Popper above). As far as cosmologists are concerned, the question is undeniably empirical, and at this point entirely open-ended; see, for example, this brief survey and the following comments from Luke Barnes (who is somewhat sympathetic to your conclusion).

No, I outlined a mapping of a possible finite past, and pointed out there are cosmological models based on a finite past (Hawking, Carroll, and Vilenkin to name 3). I am aware of no such conceptual mapping for an infinite past. — Relativist

Your "conceptual mapping" of a finite past was a semi-infinite number line. You say you cannot think of a corresponding "conceptual mapping" for an infinite past? Really?

I am sorry, but this isn't worth my time.

Yes, conceivability is subjective, but conceptions can be intersubjectively shared, analyzed, and discussed. — Relativist

Conceivability, the way you are using the word, is nothing more than an attitude, an intuition, a gut feeling. While different individuals can hold such attitudes in common, it is not the sort of concept that can be described and transmitted by a rational argument. I, for instance, do not find the beginning of time to be any easier to conceive than an infinite past, and I doubt that you could do much to change my attitude.

But then I do not make much of such attitudes. If one holds time to be an objective feature of the physical world, rather than a subjective attitude, then what does it matter if an infinite or a finite past does not sit well with one's intuitions? We are animals with a lifespan of a few tens of years; we can hardly get to grips with timespans of thousands, let alone billions of years. If we were to trust our intuitions on this, most of us would have had to be Young-Earth creationists, right? But then what are we to do with the powerful intuition that at any moment there must always be before? Or, for those having trouble conceiving of an infinite space, what are they to do with Lucretius and his spear? Intuitions just aren't a good guide to the truth in this case.

Yes, we are. On one message board that I once frequented (now defunct), which wasn't even specifically for philosophy, a subsection within its only philosophy section was created just for free will discussions.

Unlike, say, Kierkegaard's esthetics or structural realism, "free will" is the sort of subject where most people feel they can jump in without any learning or reflection. Most free will discussions are therefore trivial and confused, with people talking past each other, without even stopping to think about what free will is, or why they think of it the way they do. And I am speaking as someone whose attitude towards this subject has changed - an all too rare occurrence - from learning more about it.

It would go a long way towards making such discussions more worthwhile if participants were at least somewhat aware of the history of the subject; its relation to freedom, voluntary action, agency, autonomy, responsibility, control, determination; the role it plays in law, ethics, psychology, sociology. There is, of course, massive literature on free will in philosophy, including experimental philosophy (yes, that's a thing).

My mistake, your point is well taken. It should be said (somewhat contradicting what I said before) that even in something as seemingly dry and precise and abstract as mathematics, the actual process of coming up with a mathematical theory may start with somewhat vague, intuitive idea of what it is that you want to see, and then you evaluate your formal construction against that idea. Thus, we have intuitive, pretheoretical, or just pragmatic ideas of what a set should be, what an arithmetic should be, and then we axiomatize those ideas, giving them definite, precise form (and there can be more than one way to do that, some better, some worse, some just offering different possibilities).

But an infinite past still entails an infinite series that has been completed; that is the dilemma. Consider how we conceive an infinite future: it is an unending process of one day moving to the next: it is the incomplete process that is the potential infinity. The past entails a completed process, and it's inconceivable how an infinity can be completed. — Relativist

Well, inconceivable is a subjective assessment, it's a far cry from being provably impossible. If you just want to say that you don't believe the past can be infinite because an infinity of elapsed time seems inconceivable to you, you are welcome to it. Does an absolute beginning of time, such that right at the beginning there is no before, seem more conceivable to you?

Mathematical entities are abstractions, they have only hypothetical existence. — Relativist

That's neither here nor there, because this is true for all our thoughts, concepts, imaginings. When you think of a dog, even when the thought is prompted by looking at one, your thought is not the dog - it's an idea in your head, an abstraction of a dog.

How is this different from the infinity of mathematical operation of dividing 3 into 1? Just because it equates to an infinity of 3's after the decimal doesn't imply infinity exists in the world. — Relativist

You mean dividing 1 into 3, right? Exactly, very good example. You don't say that for there to be thirds we need to be able to write out all the decimal digits of 1/3, right? That would be an arbitrary, unjustified requirement. So why do you maintain that for there to be a "completed" infinite sequence we need to be able count out each individual element of the sequence? Does its existence somehow depend on us speaking or thinking it into existence, one element at a time? Bottom line, you can't just throw out such arbitrary requirements, you need to justify them.

Is there a theory of Absolute infinity? Please tell me if there is!!! — ssu

OK, so you make a distinction between something you call "Absolute" infinity and any other sort of infinity. I don't know what that difference is, and it doesn't look like you have a very definite idea either. When you want to find out whether something exists, you don't start by giving it a name, you start by giving it an operational definition, laying down requirements that need to be satisfied for anything to be recognized as that thing. It's no use just saying: "Well, it's Absolute, you know..."

I don't see how an instantiated infinity could ever be established empirically since we can't count to infinity. — Relativist

The same way we can empirically establish anything at all. We don't necessarily need to count to infinity for that, just as we don't need to write out all the digits of pi in order to empirically establish the harmonic oscillator solution. If a model that makes use of infinities provides a good fit for many observations, is parsimonious, productive, fits in with other successful models, etc. then we consider it to be empirically established, infinities and all.

On the other hand, I think in some cases, infinity can be ruled out. For example: the past cannot be infinite. Here's my argument:

1. It is not possible for a series formed by successive addition to be both infinite and completed.
2. The temporal series of (past) events is formed by successive addition.
3. The temporal series of past events is completed (by the present).
4. (Hence) It is not possible for the temporal series of past events to be infinite.
5. (Hence) The temporal series of past events is finite. — Relativist

"Successive addition" implies a starting point, which obviously precludes an infinite past. Your argument simply begs the question. An infinite past is a past that does not have a starting point.

I myself believe Absolute Infinity as an mathematical entity exists. It's just a personal hunch that it is so. — ssu

You don't need any hunches in order to believe that a mathematical entity exists: all you need is a mathematical theory that says that such and such entity is infinite - and such mathematics exists, there is no question about that.

It's discrete and not a continuum at all. — LD Saunders

What is?

There's no "constructing" here, space is just infinitely divisible. There's no such thing as a smallest possible distance. — MindForged

Well, actually in physics, space does not seem to be infinitely divisible. — LD Saunders

In today's physics space and time are usually modeled as a continuum. This is true for classical mechanics and quantum mechanics and for many other theories. This does not mean that we can say something definitively about the ultimate nature of space and time, or that it even makes sense to talk about such ultimate nature, as if it were uniquely defined. Conservatively, the most we can say is that current physical theories are very effective, and that gives us a good reason for thinking of space and time as a continuum and no good reason for thinking otherwise.

This doesn't mean that future physical theories will not quantize space and time. Some think that quantum physics points in that direction, although to repeat, current theory makes space and time a continuum. And an unbounded (infinite) one at that in all but some cosmological models. Speaking of which, those cosmological models with a finite or semi-infinite spacetime are so violently counterintuitive that I very much doubt that most "infinity skeptics" would be more satisfied with them than with the traditional Euclidean infinite space and time.

Dennis, if you really believe that philosophical theories are uniquely derived from experience with unassailable reasoning, and that this can be done for Aristotelian philosophy in just a couple of paragraphs, then you are very naive. Anyway, I do not wish to detail this discussion any further.

That's paradoxical. — frank

Not paradoxical, just undefined. Let's tweak the story:

- Imagine Donald Trump
- You notice he’s counting (you can tell because he is muttering and holding up his fingers). You ask how long and he says ‘I’ve been counting for ten minutes’
- What number is he on?

So put this way, this is a pretty dumb counterexample, but there are actually many puzzles involving infinities where you might think there ought to be a definite answer, but there isn't, such as Thomson's Lamp for example. There are also genuine paradoxes, where an imaginary setup that seems like it ought to be possible, in principle, leads to contradictions. But in each of these cases you have an option to reconsider your starting assumptions: Are you sure that there must be a unique answer? How do you know? Are you sure the setup itself is coherent? How do you know?

It is amazing how taste can trump analysis. — Dfpolis

Well, when it comes to philosophy, at the end of the day it does come down to "taste;" there's no getting around it, unless you believe that you can derive an entire philosophy completely a priori, without any extrarational commitments (which would be an exceptionally crankish thing to believe).

But that's not really why I don't accept your argumentation in this instance. When making an argument one must start from some common ground, and Aristotelian or Scholastic metaphysics isn't such a common ground between us. If you absolutely have to use that framework, then you would have to start by justifying that entire framework to me, or at least its relevant parts. And that is just too unwieldy a task for a forum discussion on an unrelated topic.