Saturday, March 20, 2021

Meta-abstraction in the physical and social sciences

One of the themes of Aristotle’s Revenge is the centrality of mathematical abstraction to modern scientific method, and the ways that it both affords modern physics tremendous predictive power but also, if we are not careful, is prone to generate philosophical fallacies and metaphysical illusions.  This is especially so where we are dealing with abstractions from abstractions – meta-abstractions, if you will. 

In his recent book on the philosophy of time, Raymond Tallis notes how this has happened in modern thinking about the nature of space and time.  First, physical space has come to be conflated with geometry.  Whereas the notions of a point, a line, a plane and the like were originally merely simplifying abstractions from concrete physical reality, the modern tendency has been to treat them as if they were the constituents of concrete physical reality.  But then a second stage of abstraction occurs when geometrical concepts are in turn conflated with values in a coordinate system.  Points are defined in terms of numbers, relations between points in terms of numerical intervals, length, width and depth in terms of axes originated from a point, and so on.  Time gets folded into the system by representing it with a further axis.  Creative mathematical manipulations of this doubly abstract system of representation are then taken to reveal surprising truths about the nature of the concrete space and time we actually live in.

You don’t have to be an Aristotelian to see the fallaciousness of all this.  From Berkeley to Whitehead to Lee Smolin, a diverse group of thinkers has cautioned against blithely reading off metaphysical conclusions from abstract models.  The predictive and technological successes of the abstractions have facilitated the fallacy, but it remains a fallacy all the same.  That’s why there is a longstanding debate in philosophy of science between realist, instrumentalist, and (the middle ground) structural realist interpretations of scientific theories.  The abstractions and predictive successes by themselves don’t settle anything.  Metaphysically revisionist arguments that begin “Relativity theory works, therefore…” are a bit like concluding, from the fact that you have found a certain road and highway map of Pennsylvania to be useful, that Pennsylvania must therefore really literally be nothing more than a perfectly flat white surface covered with black and red lines. 

In his book The Reformation in Economics, economist Philip Pilkington laments the ways in which the tendency to confuse what I am calling meta-abstractions with the concrete reality they are abstracted from can also infect social science and the policy recommendations based upon it.  He begins with some important points about abstraction in general.  In what follows I’ll note and expand on some of these points.

The nature of abstraction

First, following Kant, Pilkington notes that the concepts we arrive at via abstraction can have a greater or lesser degree of “homogeneity” with less abstract concepts, and with the concrete realities that fall under the concepts.  For example, the concept PLATE (as in a dinner plate) is closely homogeneous with the concept CIRCLE, as is a particular plate.  But it is less closely homogeneous with a more abstract concept like GEOMETRICAL FIGURE.  When we form concepts by abstraction, we mentally strip away concrete features of things and focus our attention on general patterns.  The more abstract the concept is, then, the more numerous are the features we are stripping away – and thus, the less there is in the way of actual concrete reality that we are capturing.  Now, Pilkington notes, the way economists represent the economy mathematically is very abstract indeed, and thus not closely homogeneous with the actual concrete economic facts. 

A second general point he makes, this time citing Berkeley, is that abstraction requires language, and, more generally, symbols.  For concepts cannot strictly be imagined.  Anything you can imagine – that is, form a mental image of – is going to be concrete rather than abstract.  To appeal to one of my stock examples, if you imagine a triangle, you are always going to form an image of some particular triangle, such as a blue right triangle.  But the concept TRIANGLE is completely universal, applying to red and green triangles as well as blue ones, acute and obtuse triangles as well as right ones, and so on.  More abstract concepts (like GEOMETRICAL FIGURE) are even further removed from anything we can form an image of.  At the same time, the way the human mind operates, we need to form some kind of image even when we entertain the most abstract of concepts, and need some kind of external sign by which we may record our thoughts about them and call the attention of others to them.  Hence we use words and other symbols.  For example, we represent the number four with the word “four,” or via the Arabic numeral 4, or the Roman numeral IV, or in stroke notation as ||||, or in some other way.

(Side note: There is a potential chicken-egg problem here insofar as nothing really counts as language in the first place – or at least, as language that goes beyond the expressive and signaling functions of which non-human animals are capable – apart from concepts.  Clearly, then, concepts and language must come about together.  But how does that happen?  Good question for another time, though for one possible answer I commend to you John Haldane’s “Prime Thinker” argument.)

Now, words and symbols are typically parts of systems of words or symbols – for example, languages, and the numeral systems just referred to – and such systems vary in their expressive power.  Pilkington cites the famous example of the advantages in mathematical reasoning that a numeral system containing 0 makes possible, compared to systems which lack any corresponding symbol.  As Berkeley writes in Alciphron, in a passage quoted by Pilkington:

But here lies the difference: the one, who understands the notation of numbers, by means thereof is able to express briefly and distinctly all the variety and degrees of number, and to perform with ease and despatch several arithmetical operations, by the help of general rules.  Of all which operations as the use in human life is very evident, so it is no less evident, that the performing them depends on the aptness of the notation… Hence the old notation by letters was more useful than words written at length: and the modern notation by figures, expressing the progression or analogy of the names by their simple places, is much preferable to that for ease and expedition, as the invention of algebraical symbols is to this, for extensive and general use.

End quote.  Now, here’s the thing.  The results we get when using a system of symbols to some extent reflect the system of symbols we’ve chosen, the things we’ve chosen to group together under a symbol, the rules that govern the system, and the ways we’ve manipulated the symbols according to those rules – rather than the objective reality being represented by the system. 

Here’s an analogy.  In a pen and ink line drawing, an artist can use thick lines to represent some contours, thin lines to represent others, a break in the line to suggest yet others, series of lines or cross-hatching to represent shadows, and so on.  Splotches of ink can also represent shadows, though they could be used instead to represent blood or holes or bumps or any number of other things.  The artist might also draw in an illustrative style or a cartoony style, do a tight rendering or a loose sketch, and so on.  Now, a skilled artist might produce a likeness of a person, object, or scene that is so close that we might find it useful for identifying the person or thing, predict how it will look from different angles, and so on.  All the same, some features of the drawing will reflect only the mode of representation rather than the thing represented, and we could be led into serious fallacies and errors if we failed to keep this in mind – for example, if we thought that the thing represented really had a black outline around it, or if we concluded that there must be some interesting relationship between shadows, blood, and holes in things, on the grounds that they all looked like black splotches in the drawing. 

Similarly, since even the most useful system of symbols will have features that reflect the natures of the symbols and the system of rules governing them rather than objective reality, we need to be careful lest we assume that there must in objective reality be something corresponding to a given element of the system.

The example of economics

In economics, Pilkington points out, one starts with abstractions like INCOME, CONSUMPTION, INVESTMENT, GOVERNMENT EXPENDITURE, EXPORTS, IMPORTS, SAVINGS and TAXES.  These already tend to run together very different phenomena.  For example, Pilkington says, CONSUMPTION can cover things as diverse as “the purchase of this book… dishwashers, bananas, underpants, blueprints for perpetual motion machines from dubious internet websites and cat-food” (p. 99).  INCOME might include Medicare payments that don’t actually go to households but rather directly to healthcare providers.  And so on. 

These abstractions are then replaced with algebraic symbols, such as Y, C, I, G, X, M, S and T, respectively, in the case of the examples cited.  These enter into formulae such as (to borrow Pilkington’s examples) the national income identity:

Y º C + I + G + (XM)

or a formula representing the relationship between income and savings and taxes:

Y º C + S + T

Such equations can then be substituted into one another, which, after cancelling, yields:

I + G + X º S + T + M

Now, Pilkington notes, we are at this point really manipulating abstractions from abstractions – what I have, again, called meta-abstractions – and thus are even farther from concrete economic reality than the abstract concepts we started out with are.  That need not be a bad thing, but one must constantly keep in mind the limits of what can be captured in such representations and the way that the results of our manipulation of formulae might reflect the method of representation rather than empirical reality.  Pilkington writes:

Such algebraic abstraction is extremely useful, but it can also be used to throw up dust and allow people to engage in sophistical nonsense.  An awful lot of microeconomics is precisely this: nice, neat, formal abstractions that have almost a zero-degree of homogeneity with the real world.  They are far, far from Kant and his circular plate.  Rather, they tend to be made-up stories with no real empirical content.  They are, in that sense, fantasy constructions. (p. 100)

Borrowing a point from fellow economist Tony Lawson, Pilkington also points out that the method of mathematical modelling is such that the model is made into a closed system, with the result that the reality modeled is represented in a deterministic way.  This may be appropriate in physics, Pilkington says, but not when representing human action.  To treat economic behavior as if it were deterministic, even just for purposes of the model, is a clear case of reading what is really just a feature of the abstract method of representation into the reality represented, rather than reading it out of that concrete reality.

(I would add, though, that even in physics we should regard such models as simplifying abstractions, for reasons of the sort raised by Nancy Cartwright.  Of course, these days people are, in light of quantum mechanics, happy to allow that nature does not operate in a strictly deterministic way.  But even in the old days, the contrary supposition was not really justified.  The determinism was read out of the mathematical models and into nature, not out of nature itself.  Again, see Aristotle’s Revenge for discussion of such issues.) 

The bias in science

Now, here is a deep irony.  Mathematics is, quite rightly, widely regarded as a paradigm of objective and disinterested knowledge.  And its heavy use in modern science is a major reason why science too has a reputation for objectivity and disinterestedness.  But there is a fallacy hidden in the implicit inference.  For the fact that a mathematical technique is of itself free of bias simply does not entail that its application to some aspect of concrete reality is also free of bias.  And indeed it is not free of it.  When we apply mathematical models to objective reality, we are always making the tendentious assumption that there is nothing more to reality than is captured by the model – or at least nothing more to it that is relevant to the purposes for which we are constructing the model.  Of course, that assumption might be defensible and correct; but then again, it might not be.  Either way, it is an assumption, and one that is extrinsic to science itself.  It is a philosophical assumption about science, an assumption that reflects further philosophical assumptions about what nature is like and about what the best techniques are for studying it.  As E. A. Burtt noted in his classic book, the tendency of many modern scientists has been to make an entire metaphysics out of what is really just a method. 

Pilkington too complains that scientists tend to make philosophical assumptions about which they are “completely unreflective” (p. 114).  For example, they “have a tendency to fall back on a reactive materialism as their default worldview” but “the details are never worked out – and if we are to be honest, it more so resembles an ideology than a properly reasoned and considered worldview” (pp. 113-14).  He also notes that, even when making grand but undefended philosophical assumptions, “most scientists arguably do not truly understand the philosophical implications of what they their theories tell them because they are so illiterate in the language of philosophy” (p. 114). 

Pilkington observes that though what he calls “lower-level physics” such as “Newtonian mechanics, electromagnetism, thermodynamics, the basic precepts of relativity theory and so on” are relatively free of bias, “speculations about the origins or nature of the universe” are not free of it and often verge on “crossing the boundary into metaphysics” (p. 121).  The former areas of study have a degree of empirical testability that the latter do not, but because they are all lumped together as “physics,” the latter inherits from the former an unearned prestige.

Then, as Pilkington notes, there is the fact that a branch of science like “Newtonian physics is almost completely bereft of politicisation because, say, the theory of gravity really makes no difference to how we as human beings organise our lives and build our societies” (p. 117).  However, when scientific study does have dramatic political implications, the evidence is clear that it is often massively biased.  (Pilkington cites the work of Brian Martin, who took as a case study the debate several decades ago over the effect of supersonic transport aircraft on the ozone layer, and showed how politicized the science on both sides had become.)

Now, consider these four sources of potential bias in science: an excessive confidence in abstract mathematical models; a tendency toward a crudely materialistic analysis; a fallacious attribution of the prestige enjoyed by the directly testable areas of science to the untestable and speculative areas; and a subject matter that has dramatic implications for how we organize our lives and societies.  Have we seen these come together in any recent controversies?   Hmm?

Well, of course we have.  All four have been on display in the defense of the COVID-19 lockdowns, which involved an overreliance on speculative and faulty mathematical models; a fixation on the mechanics of the transmission of the virus while ignoring the psychological damage, destruction of livelihoods, and ruin to education caused by lockdowns; the ridiculous smearing of all skeptics as “science deniers,” as if questioning the lockdown was on all fours with rejecting the Periodic Table of Elements; and considerable politicization, putting vulgar sloganeering in place of dispassionate scientific argumentation and favoring more relaxed measures for political allies such as left-wing protesters.

Of course, opponents of lockdowns can also be influenced by political biases, no less than proponents of lockdowns can be.  That is precisely why, a year ago and early in the pandemic, I was defending the proponents against those who too quickly dismissed the initial lockdown as an unjustifiable infringement on liberty.  However, in the nature of the case it has always been the defenders of lockdowns, and not the opponents, who have to meet the burden of proof.  And as I also argued a year ago, the longer the lockdowns went on, that burden would inevitably become heavier rather than lighter. 

Yet as time went on, lockdown defenders largely acted as if the opposite were the case.  Indeed, they largely conducted themselves disgracefully, exhibiting precisely the reverse of the caution and humility that the four sources of potential bias cited by Pilkington should have made them especially sensitive to.  And at this point it is clear that the most draconian measures were a mistake, inflicting massive harms with no net benefits that could not have been achieved through less extreme measures.  But “mistake” is really far too mild a term for the callous arrogance and manifestly fallacious reasoning of people who imposed on others enormous costs that they mostly avoided themselves.  The dogmatic scientism that motivated this disaster in policy provides an object lesson in how philosophical errors are by no means of mere academic interest, but can have dramatic and indeed catastrophic real-world effects.

Related posts:

Concretizing the abstract

David Foster Wallace on abstraction

Cundy on relativity and the A-theory of time

Color holds and quantum theory

The particle collection that fancied itself a physicist

Think, McFly, think!

Metaphysical taxidermy


  1. The biggest hoax in the history of science is that an abstract "objects" like time can dilate. See Wolfgang smiths latest book (Physics and Vertical Causation: The End of Quantum Reality) and (Teilhardism And The New Religion: ).

    The moment you realize relativity was created to attack the Catholic church, your eyes will be wide open.

    I know smart Catholics like 'grodrigues' has fallen for these thing. That why Saint Paul says that even the elect will fall for the upcoming deception.

    1. The claim that Einstein created relativity to attack the Catholic Church ranks as one of the most nutjob ideas ever to appear on this site, and these are legion.

    2. That why Saint Paul says that even the elect will fall for the upcoming deception.

      Where does Paul say this?

    3. Relativity being created to attack Kant would be more plausible. Dude screwed up german philosophy and Einstein was a german that liked philosiphy.

  2. On pretending that abstractions like time can be treated like a concrete object instead of an abstraction and that it can dilate, see also Wolfgang Smiths: "Ancient Wisdom and Modern Misconceptions: A Critique of Contemporary Scientism"

  3. "I know smart Catholics like 'grodrigues' has fallen for these thing. That why Saint Paul says that even the elect will fall for the upcoming deception."

    Catholic I accept because I am, but there was no need to hurl the "smart" insult, that's just mean.

    1. I honestly would like your opinion on Wolfgang Smiths latest tomes. I don't know how familiar you are with them, but I also don't know if you have a blog. Just a review of those from your perspective would be awesome.

    2. @grodrigues, Since you're a mathemtician, I'd like to ask you a question about Nonmeasurable sets that's relevant to metaphysics:

      Since Nonmeasurable sets have no defined value, one possibility from this could be that brute facts are a type of Nonmeasurability with regards to probability. And if Nonmeasurable sets are possible, then brute facts could be possible too.

      And in fact, Alex Pruss uses the example of throwing a perfectly sharp dart uniformly at a Nonmeasurable target as an example of how some Nonmeasurable object can lead to a lack of any meaningful probability, just like brute facts.

      Now, in your opinion, is this connection between Nonmeasurability and brute facts accurate? Does the possible existence of Nonmeasurable sets imply something could also have no meaningful probability as well, say via dart-throwing at a Nonmeasurable target?

      Now one possible response to this could be to point out that Nonmeasurable things necessarily require the Axiom of Choice, which then implies non-constructibility of any Nonmeasurable set - or at least that there is no effective explicit procedure to form such a set, and so this rules out any concrete causal instantiation in any world for them.

      Another important note is that even if we accept the real possibility of Nonmeasurable mathematical constructs, such constructs aren't absolutely nonmeasurable like brute facts would be, but still have probability bounds such as this:

      P of hitting the nonmeasurable target < P of throwing it < 1.

      So there is a probabilistic context which reduces the amount of skeptical worries or unlivable consequences of such Nonmeasurable constructs.

      What do you think?

    3. @JoeD:

      "Since Nonmeasurable sets have no defined value, one possibility from this could be that brute facts are a type of Nonmeasurability with regards to probability. And if Nonmeasurable sets are possible, then brute facts could be possible too."

      Stop right here. First, recall the mathematical facts: on ZFC it is provable that there exist non-measurable sets, that is, a subset of R (to fix ideas) that has no defined measure. One such construction is Vitali's construction, but it relies on the axiom of choice crucially; in fact all such constructions rely on it, because there are models of ZF + DC (DC is dependent choice) where *every* subset of R is measurable, so there can be no effective procedure to spit out such a set or even a "reasonably effective" description of such sets (making this more precise needs heavy mathematical logic). Then there are theorems to the effect that non-measurable sets are pretty much unavoidable, like the Banach-Tarski paradox, which like all great mathematical ideas, lead to entire mathematical theories, in this case the theory of amenable groups.

      With that out of the way, I do not understand why you say "if Nonmeasurable sets are possible, then brute facts could be possible too". What is the connection between one thing and the other? I do not even know exactly what you mean by "Nonmeasurable sets are possible". A set is a mathematical object. What does "possible" mean in this context? Finite sets have a pretty straightforward relation with the objects of our ordinary experience. Infinite countable sets already break its bounds. Infinite, uncountable sets (non-measurable sets are necessarily uncountable)? Dragons be here.

    4. @grodrigues,

      Well, the example I gave explains the connection between brute facts and Nonmeasurable sets:

      If we decide to hit a nonmeasurable target with a dart - or hit a target that has a maxially nonmeasurable subset S, the probability of hitting it would lack any meaningful value.

      Which is exactly the same kind of probability - or lack thereof - of brute facts. That is, brute facts are nonmeasurable with respect to probability of them happening - it's the concept of nonmeasurability applied to probability for something being actualised.

      As for what it means for Nonmeasurable sets to be possible - I was referring to their possible existence. If the concept of Nonmeasurability or Nonmeasurable sets is coherent, and can be proven with the Axiom of Choice which itself is coherent, then Nonmeasurable sets seem to at least be logically possible. So at least God could create them.

      And if Nonmeasurable sets are truly metaphysically possible and actualisable, and if one of the consequences of a particular set-up (namely hitting a nonmeasurable target) is a lack of any meaningful probability, then this is the same as for brute facts which lack probability. So brute fact would just be a particular subset of Nonmeasurability with regards to probability.

      Now what do you think of the above reasoning?

      And specifically, what do you think of the responses given to it - that Nonmeasurable sets are non-constructible and so can't be causally actualised, so can't exist in a causal context? Or that even if we admit their possible physical existence, that they still wouldn't be as absolutely nonmeasurable as brute facts due to the probabilistic bounds cited?

    5. @Tap:

      "I honestly would like your opinion on Wolfgang Smiths latest tomes. I don't know how familiar you are with them, but I also don't know if you have a blog. Just a review of those from your perspective would be awesome."

      Several years ago, I got myself a copy of Smith's first book "The Quantum Enigma", but still have not found the time to read it. And since I am much more interested in QM than in GR (time dilation already appears in SR, but I classify it as a special case of GR), the odds are that I will never read the book you mentioned.

    6. @grodrigues, Another quick side question or two, unrelated to brute facts and instead about constructibility and partition as it relates to God:

      1) Since the Banach-Tarski paradox crucially depends on there being Nonmeasurable subsets of a sphere, is the reason why the duplication can't be constructed in both technical mathematical terms and practical everyday ones precisely because Nonmeasurable things can't be handled due to being Nonmeasurable - they can't be picked or chosen since that requires determining where they are or containing them, which is impossible by definition of them lacking any meaningful site? And maybe also that Nonmeasurable things can't be handled on a step-by-step basis, which is required for construction?

      Or is B-T non-constructible for other reasons - say the Axiom of Choice by its definition of choice or the way it is operated doesn't allow constructibility by definition?

      2) And even if the B-T partition were non-constructible due to either of these reasons or other ones - providing Nonmeasurable sets can also actually exist in reality would this mean that God could perform the B-T construction by directly actualising it without any constructive method?

      Say, just as the B-T partition can still be "done" in a way with our intellect, since our mind abstractly goes through the steps without actually acting on anything physical?

      That is, the B-T can only be done by directly actualising the steps without any physical consruction - which is something God could do by omnipotence?

    7. @JoeD:

      "If we decide to hit a nonmeasurable target with a dart - or hit a target that has a maxially nonmeasurable subset S, the probability of hitting it would lack any meaningful value."

      I am not trying to be hard here, but what is this set that we are supposed to imagine hitting with a dart?

      Here is a possible way forward: the usual source of such non-measurable sets is that we model space-time as a manifold, and we model manifolds as sets of points with a topology, etc. and etc. so a would-be such set would be a set of points. Here is the problem: manifolds as sets of points is an artefact of a specific formalization. It is perfectly possible to do geometry with "pointless" manifolds, e.g. as in Synthetic Differential Geometry -- to be fair, I am not sure if anyone has actually gone to the trouble of doing it, but I am pretty sure it is possible -- and in such models you (usually) do not have AC available, so neither can you construct non-measurable sets. I would be *seriously* surprised if the underlying physics depended on such details of the specific formalization.

      So if I understand you right, by "possible" what you mean is the actual existence of a non-measurable set of "things". But here we face the first hurdle: what are these "things" that we are suppose to aggregate by an act of the understanding into a set? If you answer substances, Aristotle's arguments against actual infinite sets, seem to me to be dispositive. If you mean things like points, then the real question is the metaphysical status of points and sets thereof. As I have said above they are certainly not essential as far as physics goes.

      To finish these (long, too long) remarks, I am not exactly phazed by the apparent weirdness of probability outcomes with infinite sets. Infinite sets are weird, marvellously so.

      note: I will answer the questions on B-T latter.

    8. @grodrigues,

      I'm not talking about the physics of our world at all, but the possibility of nonmeasurable things (here on NS means both Nonmeasurability as a trait or property, and also a set depending on the context).

      The argument is essentially that since math shows that NS's are coherent and can even be proven to exist in math assuming the equally coherent AoC, then at the very least God could create NS's in the real world since NS's are proven to be a possible concept from math.

      And if NS's really are the sort of thing that could exist at least by divine power, then this clearly implies a possible situation where, for example, a dart could be thrown at a NS target, which necessarily lacks any meaningful probability.

      And it just so happens that another thing that lacks any probability are brute facts - so brute facts really seem to be just a subset of NS applied to probability. A brute fact is something being real or becoming real for no explanation or cause - which parallels the lack of meaningful value of the size of sizeless Vitale sets or the lack of probability of Throwing-Darts-at-NS.

      So if NS's can actually exist - since they are coherent and possible and even provable in a sector of math - then it seems brute facts can also since they are just Nonmeasurability applied to things happening or being actual in terms of their lack of explanation.

      As for what those NS things are; Aristotle's argument against infinite substances may only apply to generating an infinite set from finite causes, but God could still directly create an infinite set by His omnipotence so that doesn't disprove an infinite set of substances that could be aggregated. As for the NS set being a set of points - I don't know much about the metaphysical status of points to say either way, but the possibility of NS as a trait seems to be independent of that question.

      So it is in this context of NS being possible - nonmeasurability as a trait or property, or sets specifically, either works - that the other arguments I mentioned are brought up.

      The first point is that even if NS as a either a trait of any thing or a set is truly logically possible and could exist, it is by definition non-constructible due to Choice, so mathematical constructs that are NS can't be actually causally implemented in concrete reality due to that.

      The second point is that even if NS constructions could be actually causally implemented in reality, there would still be a difference between them and brute facts:

      Brute facts would be absolutely nonmeasurable or NS while NS constructs such as sets or targets would still have probabilistic bounds such as P of hitting the NS being lesser than the P of throwing the dart, which in turn is lesser than 1.

      What do you think of these arguments that if NS is truly possible and could exist then brute facts could also truly exist especially if they are a special subset of NS applied to probability of being actual? As well as the two points in response?

    9. @grodrigues, Note about NS possibly existing in the real world - what I mean is not necessarily that we should model physics in a certain way or that the physics of our world naturally allows NS, but that God could create a different world from ours - say a possible world where NS do exist, since NS's seem possible.

      It's a matter of possibility and possible existence in some world - and of course, I'm not using Possible Worlds of analytic philosophy here, just to clarify.

    10. @JoeD:

      On the B-T questions:

      (1) The sets in the paradoxical decomposition are necessarily non-measurable, hence the existence of such decompositions needs AC (actually, less than full AC is needed but still more than DC, which is often taken as the constructive core of classical mathematics). This is a purely mathematical statement. Non-constructibility follows, where non-constructibility is understood in a *mathematical* sense, because if it were constructive such construction could have already been done in ZF + DC. There are a few things more that could be said, but this entails having precise descriptions of what counts as a construction. To illustrate the potential problems, transfinite recursion constructions could maybe satisfy your step-by-step requirement. Sometimes they need AC (to first set up the chain of ordinals over which you recurse), but sometimes they don't.

      (2) You ask "Nonmeasurable sets can also actually exist in reality would this mean that God could perform the B-T construction by directly actualising it without any constructive method?" This is way, way beyond my pay grade. I think we have little to no grasp of what God could possibly do or not, in the same sense we have little grasp on His essence, and I do not think mathematics provides any special insights on the question, as I do not take mathematical existence (as a theorem of some formal system or other) as any sort of reliable guide, or even mere logical consistency as a sufficient requirement.

    11. @JoeD:

      "I'm not talking about the physics of our world at all, but the possibility of nonmeasurable things"

      Yes, but my problem is this: I know what a non-measurable set is, mathematically speaking. In ZFC, it is provable that they exist. But how do you go from a mathematical existence proof to possible existence? That is why I focused on the physics and its formalization: you have to tell me what it is that you mean that "possibly, a non-measurable set of 'things' exists" as whatever problem you think there is may be in the 'things', it may be in forming sets, or it maybe in the structure over and above sets as to speak of non-measurability you have to have a measure around so that the 'things' you are aggregating cannot just be a random pile of things. Brushing this all aside by going from a mathematical existence proof to possibly, God could have created a world in which such a set of things exists is an unwarranted leap.

      "And if NS's really are the sort of thing that could exist at least by divine power, then this clearly implies a possible situation where, for example, a dart could be thrown at a NS target, which necessarily lacks any meaningful probability."

      This again is an unwarranted leap. Even if there is some sense in which one can say that non-measurable sets exist, what exactly is the problem that they don't have a well-defined probability in some contrived scenario or other? I could be really finicky and ask are darts with point-like tips metaphysically possible?

      I have never seen, touched or stubbbed my toe in a set. Sets are abstractions, formed in the mind, and the usual deflationary account of them is to say that they are nothing over and above their individual elements. So in front of me I have pen and a book, so I can form the set {pen, book}, but to say that this set exists entails nothing above and beyond saying that the pen and the book exist. The same for any sets that could possibly exist. All this to repeat my point: you have to do way more work to derive any metaphysical conclusions from any piece of individual mathematics, and in the course of doing it, it is what metaphysics you slip in that will be doing the real work, not the mathematics.

    12. I only have a little bit to add to what grodriguez already said. It's not only that unmeasurable sets are abstract constructs that we don't know how to physically manifest, but also that they would defy any sort of physical manifestation. To do so would require controlling amounts of material that go below the Planck length.

    13. @grodrigues,

      "Non-constructibility follows, where non-constructibility is understood in a *mathematical* sense, because if it were constructive such construction could have already been done in ZF + DC. There are a few things more that could be said, but this entails having precise descriptions of what counts as a construction. "

      But didn't you also say when describing the Banach-Tarski paradox in a different post that the non-constructibility also implies a few practical senses of the term that aren't technical?:

      "As to the Banach-Tarski it is probably worth it to mention that its proof is unavoidably non-constructive -- non-constructive in a technical mathematical sense, but that does translate in a few more "practical" senses, such as there being no effective constructive procedure for the partition of the ball, etc."

      From here:

    14. @JoeD:

      In the two passages, I am using practical in different senses,m as witnessed by the fact that in the second passage I enclose the word in square quotes and lead with "such as there being no effective constructive procedure for the partition of the ball" -- "practical" here means "interesting to mathematicians". Yeah, the word choice is suboptimal to say the least...

    15. @grodrigues,

      So... could we still say that this implies it can't be causally implemented or physically constructed - especially since physical constructibility seems like a lower level definition of constructibility that is even more limited than mathematical constructibility?

    16. Just barging in, but

      "So if NS's can actually exist - since they are coherent and possible and even provable in a sector of math - then it seems brute facts can also since they are just Nonmeasurability applied to things happening or being actual in terms of their lack of explanation."

      This is not what brute facts are. Pruss argues that brute facts would, if possible, have no meaningful objective probability, but the idea that if NS are possible then brute facts are possible is nonsense; brute facts are absolutely unexplained contingent events and there is nothing to properly constrain them since they are supposed to come from nothing, by nothing. There is no process of selection through any probabilities; there is nothing that selects among any possible worlds, the brute facts are not caused, there would be nothing selecting or hitting anything.

    17. @JoeD:

      "So... could we still say that this implies it can't be causally implemented or physically constructed - especially since physical constructibility seems like a lower level definition of constructibility that is even more limited than mathematical constructibility?"

      Could God have created a world in which the physical laws are such that in some sense, non-measurable sets can be instantiated? I don't know and I don't know how one would even begin to try to answer this question. Assuming it is possible, does that entail any skeptical worries? I have still not seen an argument for why that is so. In your contrived scenario, enclose the non-measurable set by an appropriate figure of area/volume/whatever 1 (1 what? choose whatever unit you fancy). Then the probability that a point-tipped dart hits the set is undefined. Why is that a problem? Don't know. In the frequentist interpretation, the probability is a limit of a certain ratio. Why is the non-existence of this limit somehow problematic? Shrug shoulders.

    18. @Unknown,

      "There is no process of selection through any probabilities; there is nothing that selects among any possible worlds, the brute facts are not caused, there would be nothing selecting or hitting anything."

      Are you referring to the argument that hitting a NS target with a dart still has probability bounds - P of hitting the target being less than P of throwing the dart, which is in turn less than certain truth or 1?

      Or are you saying that the Axiom of Choice - which is used to select elements in cases where there are no proper distinguishing features and an infinity of elements - by its very nature as a choosing function means it's NOT arbitrary as brute facts have to be?

      If the latter, that certainly seems plausible - though when I asked Alex Pruss about NS and brute facts, he did say that NS sets would lead to less skeptical worries than brute facts due to probabilistic bounds on NS constructions - the keyword being less skeptical worries, so some skeptical worries seem to still exist with NS.

      Pruss also compared NS to brute facts in his paper on probability in science and PSR - arguing how if brute facts are possible then the probability of them occurring is nonmeasurable and thus undermines science, so we need an a priori way to get rid of brute facts.

      He also compares the idea of throwing a dart at a NS target to brute facts in that they are both defeaters for any inductive expectation - the lack of meaningful probability for the dart hitting the NS target seemingly the same as the nonmeasurability of brute facts.

      Plus, nonmeasurable sets are nonmeasurable because they lack any meaningful size - they are literally called sizeless sets since their size is neither 0 not infinite - so size does not apply to them at all. They are just sets that exist, and happen to have no size.

      In the same way, brute facts would also seem to be an instance of NS since the concept of probability doesn't apply to them at all - they are literally probability-less. They are things that can happen, without probability.

      What do you think?

    19. @grodrigues,

      About probability - the reason why the connection is made to brute facts is that brute facts also have an undefined probability. If nonmeasurability is basically being undefined, and nonmeasurable sets with no defined size can exist, and those same sets in some contexts imply undefined probability which is the same as for brute facts, then the possibility of nonmeasurable sets existing lends more credibility to brute facts being possible as well. At least that's the argument.

      To clarify the scenario - it's specifically that the Probability of hitting the nonmeasurable target is LESSER than the Probability of throwing the dart which is in turn LESSER than 1 or absolute certainty. This is in fact likely a DIFFERENCE between nonmeasurable sets and brute facts since brute facts don't have any probability bounds while this scenario at least does.

    20. @Unknown,

      One more thing came to mind - the Axiom of Choice generates nonmeasurable sets from selection, but that itself can be said to be something we have to do to discover NS sets, not that it creates them.

      For example, if a nonmeasurable set were created ex nihilo - either by God or if that happened as a brute fact - it would have members that we would usually generate via particular selection, but in itself the recently created NS set wasn't created by selection; rather, all of its members are part of its definition or internal structure which came into being with it.

      So it seems the selection of sets via AoC doesn't constrain them from being brute if they popped into existence rather than being generated via a method.

    21. @Unknown,

      And couldn't one describe brute facts as a type of selection as well? It may not be caused or explained, but a particular brute fact is distinct from other brute facts, so when something specific happens brutely it was in a sense selected since this particular thing happened for no reason, which is by definition distinct from another thing.

      So you could say nothing selected it, rather than denying that selection applies to brute facts or that we cannot speak of selection in any way regarding brute facts.

    22. @JoeD:

      "If nonmeasurability is basically being undefined, and nonmeasurable sets with no defined size can exist, and those same sets in some contexts imply undefined probability which is the same as for brute facts, then the possibility of nonmeasurable sets existing lends more credibility to brute facts being possible as well. At least that's the argument."

      Non-measurability is not being measurable, or not having its measure defined, with the context specifying what measure we are talking about. I am not being pedantic for the sake of pedantry: Solovay showed that the existence of measurable cardinals (a statement independent of ZFC) is equiconsistent with the existence of a sigma-additive extension of the Lebesgue measure to *all* subsets of the real line (this is not inconsistent with Banach-Tarski type results as these rely on the measure being invariant over a non-amenable group such as the group of rotations in R^3). The connection of "Undefined probability" to "brute facts" is baffling, meaning, I can see why brute facts do not have causes (or explanations -- I tend to view all explanations as causal), not even probabilistic ones, but not the reverse and certainly not "the same".

      "To clarify the scenario - it's specifically that the Probability of hitting the nonmeasurable target is LESSER than the Probability of throwing the dart which is in turn LESSER than 1 or absolute certainty. This is in fact likely a DIFFERENCE between nonmeasurable sets and brute facts since brute facts don't have any probability bounds while this scenario at least does."

      The probability of hitting the non-measurable set is undefined so it is not lesser or greater than anything whatsoever.

    23. @grodrigues,

      Part 1 of 2

      1) Well the context in this case is about probability of occurrence, which is the measure that lacks definition. And it seems straightforward as to how undefined probability is connected to brute facts - because brute facts have undefined probability.

      Their lack of any causes or explanations intrinsically is what entails no defined probability.

      2) Well, here is how Alex Pruss put it:

      "I think there is some reason independent of the PSR to doubt the actual causal implementability of nonmeasurable mathematical constructs. Cf. the chapter on the Axiom of Choice in my book on the paradoxes of infinity."

      "I also think there is a difference between (a) the extreme nonmeasurability of uncaused contingent events and (b) the kind of nonmeasurability you would have in a concrete situation like throwing a dart at a nonmeasurable target. In (a), there are no probabilities at all attachable to the event. But in (b), there are probabilistic bounds, since the probability of hitting the nonmeasurable target is less than the probability of throwing the dart which in turn is less than one. Thus, (b) leads to less in the way of sceptical worries than (a)."

      And here is a paper by Pruss about PSR and probability:

      He basically applies Bayes to nonmeasurable events - which is what brute facts are supposed to be.

      As for the 17-page article, here are some extracts about the dart throwing example and why brute facts are mathematically nonmeasurable things:

      "So suppose coins are being flipped in this roundabout way. A dart with a perfectly defined tip (say, infinitely sharp or perfectly symmetrical) is uniformly randomly thrown at a circular target. There is a region A of the target marked off, and a detector generates a heads toss (maybe in a very physical way: it picks up the coin and places it heads up) whenever the dart lands in A; otherwise, it generates a tails toss."

      "The chance of the coin landing heads now should be equal to the proportion of the area of the target lying within A, and we can elaborate Cp to the hypothesis that the area of A is pT, where T is the area of the whole target. Given the observation of approximately 750 heads, it seems reasonable to infer that probably the area of A is approximately (3/4)T. But what if an area cannot be assigned to the marked region A? Famously, given the Axiom of Choice, there are sets that have no area—not in the sense that they have zero area (like the empty set, a singleton or a line-segment), but in the sense that our standard Lebesgue area measure cannot assign them any area, not even zero. In such a case, there will also be no well-defined chance of the dart hitting A, and hence no well-defined chance of heads. Let N be the hypothesis that A has no area, i.e., is non-measurable."

      "Now, here is a fascinating question. If N were true, what would we be likely to observe? Of course, if we perform 1000 tries, we will get some number n of heads, and n/1000 will then be a frequency between 0 and 1. We might now think as follows. This frequency can equally well be any of the 1001 different numbers in the sequence 0.000, 0.001, . . . , 0.999, 1.000. It seems unlikely, then, that it’s going to be near 0.750, and so C3/4 (and its neighbors) is still the best hypothesis given that the actual frequency is 0.750."

    24. @grodrigues,

      Part 2 of 2

      "But this reasoning is mistaken. To see this, we need to sharpen our hypothesis N a little more. There are non-measurable sets A where we can say things about the frequency with which A will be hit. For instance, it could be that although A is non-measurable, it contains a measurable set A1 of area (0.74)T and is contained in a measurable set A2 of area (0.76)T. (Think of A as 74% of the target plus a nonmeasurable set localized to an area containing only 2% of the target.) But the dart will hit A1 about 74% of the time, and A2 about 76% of the time. Whenever the dart hits A1, it hits A, and whenever it hits A, it hits A2, so we would expect the dart to hit A approximately 74% to 76% of the time. And so our observed frequency would be no surprise. The mere fact that A is nonmeasurable does not rule out probabilistic predictions about frequencies, because a nonmeasurable set might be “quite close” to measurable sets like A1 and A2 that bracket A from below and above."

      "However some sets are not only nonmeasurable, but saturated nonmeasurable. Set A is saturated nonmeasurable provided that all of A’s measurable subsets have measure zero and all measurable subsets of the complement of A are also of measure zero. Given the Axiom of Choice, for Lebesgue measure on the real line, there not only are nonmeasurable sets, but there are saturated nonmeasurable sets (Halperin 1951). When A is saturated nonmeasurable, no method of generating predictions by bracketing a the probabilities into an interval, like the one from 74% to 76%, will work. The only measurable subsets of A will have zero area and the only measurable supersets of A will have area T. So our bracketing will only tell us the trivial fact that the frequency will be between 0 and 1, inclusive."

      "Let M then be the hypothesis that A is saturated nonmeasurable. Can we say that given M, the frequency is unlikely to be near 0.750?
      The answer turns out to be negative even if we have infinitely many observations. Pruss (2013) has given a plausible mathematical model of an infinite sequence of independent identically distributed saturated nonmeasurable random variables, and it follows from his Theorem 1.3 that for any nonempty interval I which is a proper subset of [0, 1], the event that the limiting frequency of the events is in I is itself saturated nonmeasurable..."

      "So in our case above, if I is some small interval like [0.740, 0.760] centered on 0.750, there is nothing probabilistic we can say about the observed frequency being in I when the target set is nonmeasurable as the event of the frequency being in I is then saturated nonmeasurable. In particular, we cannot say that the frequency is unlikely to be near 0.750 (nor that it’s likely to be near). The observation of a frequency close to 0.750 is neither surprising nor to be expected given that the target set is saturated nonmeasurable."

      "Our probabilistic reasoning thus cannot disconfirm hypotheses of saturated nonmeasurability. Such hypotheses endanger all our local scientific inferences from observed frequencies to chancy dispositions, inferences central to our epistemic practices. Yet our local scientific inferences are, surely, good. If we cannot disconfirm saturated nonmeasurability hypotheses a posteriori, then we need to do so a priori."

      What do you think of this?

    25. @JoeD:

      "Their lack of any causes or explanations intrinsically is what entails no defined probability."

      You have made my point for me. To repeat, I can understand brute facts have no explanations, not even probabilistic ones, but if there is some event or other for which there is no defined probability (leaving aside all the care needed to make this rigorous), there is no reason to start panicking as if the impending doom of brute facts is upon us. It just follows that such a scenario has this in common with brute facts: it has no defined probability.

      About the 2), I understand and I concur with Pruss's remark from the qualitative difference between brute facts and events, to call it that, with undefined probability. It is another way of stating my above point. I do *not* understand the argument he seems to be advancing about probability bounds. He seems to be saying that 0 <= P(E) <= 1 but P(E) is undefined -- so what the heck is he saying?

      Then he continues by assuming that there are measurable sets F, G such that F <= E <= G from which he "derives" the probability bound P(F) <= P(E) <= P(G). Same point as before: P(E) is undefined so the inequality P(F) <= P(E) <= P(G) is non-sensical. It *seems* as though if such a scenario is possible that we would have such a probability bound, but I repeat, if P(E) is undefined what is the meaning of such would-be bound? The limit of the ratios of hits to throws does *not* exist so what are we talking about when we are talking about P(E) and would-be bounds? I suppose I would have to look up his 2013 paper, mentioned at the second paragraph from the end, after Halperin's result on the existence of saturated non-measurable sets. In it he develops a "plausible mathematical model of an infinite sequence of independent identically distributed saturated nonmeasurable random variables", so I guess it depends on what "plausible" means here.

      In summary, I have no problem with his final conclusion of a priori ruling out non-measurable sets. I just think the argument can be shortened to "if they could be instantiated they would have undefined probability (in some scenario or other) and it could not be checked, confirmed or disconfirmed" and leave it at that. In other words, the argument for disallowing them is an epistemological one, and it only impinges on the actual world (the only one that matters, by the way). Whether God could create a world in which such sets are instantiated is a different question altogether.

    26. @grodrigues,

      1) You're right it doesn't strictly prove brute facts exist, but it can be used to make brute facts more likely and argue for their probable existence, since one of the main problematic things about BFs is their lack of defined probability which implies skepticism and agnosticism.

      Especially so if brute facts are just a special type or subset of nonmeasurability, or if it's just the universal concept of Nonmeasurability applied to causation or explanation.

      2) The paper in question would be “On the Law of Large Numbers for nonmeasurable identically distributed random variables”, so if you want to read it you can look for that.

      3) One more thing - could one argue that mathematical nonmeasurable sets are radically unlike brute facts in that they are selected and have a certain order to them?

      That is, in order to create a NS set one has to use the Axiom of Choice and actually choose particular members of other sets to do that. The existence of NS depends on choosing in contexts where there is an infinite amount of things and no distinguishing features for otherwise natural choice functions - and that choice isn't a brute fact in itself.

      Heck, even if a Nonmeasurable Set or Object wasn't constructed by conscious and rational humans but by machines it would have this trait.

      The only way to get rid of this internal order would seem to make the NS set pop into existence out of nothing with no procedure of generating it by choice.

      What do you think?

    27. @JoeD:

      "Especially so if brute facts are just a special type or subset of nonmeasurability, or if it's just the universal concept of Nonmeasurability applied to causation or explanation."

      Non-measurability is a property that subsets of a measure space have. I have no idea what "universality" you are talking about, certainly not when connected to causality. Brute facts are contingent facts, states of affairs, events, whatever that have no reason, raison d'être, explanation, etc. They are distinct things.

      "That is, in order to create a NS set one has to use the Axiom of Choice and actually choose particular members of other sets to do that."

      To show that non-measurable sets exist one has to use some (relatively strong) choice principle, in the exact same sense that to show that infinite sets exist one has toi use the axiom of infinite (e.g. there are no infinite sets in the topos of finite sets, or there are no non-measurable sets in ZF + DC + LM). If a set is non-measurable it is non-measurable and that is it, that is, non-measurability is a property like cardinality or other set-like properties. In the mathematical context, it is so necessarily and that is it, that is the whole explanation. Assuming non-measurable sets could be instantiated, the causal explanation needed/lacking would be of their origin, not in their being non-measurable. If their production and coming to be relied on some divine fiat involving a more or less concrete application of AC then that would be the explanation. An analogy: electrons have spin 1/2, because necessarily electrons have spin 1/2. The explanation needed/lacking would be of their origin or in other contingent properties like their state, energy, momentum or what have you.

    28. @grodrigues,

      Understood, though since the creation of a non-measurable set would have to involve a concrete application of AC which isn't undefined itself, this would be one of the major distinguishing features of NS over brute facts. The very principle of AC isn't unintelligible and has an order to it - is that correct?

      Also, since one uses AC to create a non-measurable set by bringing together members from other things, could one then create different non-measurable sets based on what "elements" are brought to the table or how they are "arranged"?

      If so, this crucial element of how they would be created via AC seems to bring some intelligibility to them which helps them make sense.

    29. @grodrigues,

      Also, here's an argument Alex Pruss uses to justify the idea that non-measurable constructs still have probability bounds:

      "Imagine you're randomly throwing a dart at a round target C. Let A be a nonmeasurable subset of C. Suppose you also have a fair coin.

      Now, let E be this event: you hit A with the dart AND your coin lands heads. Then the probability of E is less than or equal to the probability of heads, which is 1/2. So we have an upper bound on E. Yet E is nonmeasurable.

      Similarly, we can get lower bounds. Let F be: you hit A with the dart OR your coin lands heads. Then the probability of F is at least 1/2."

      What do you think of that?

  4. The microeconomics part made me remember the austrian school, and praxeology in particular. The normal use of it that i see consist in a bunch of logical deductions while taking for granted that reality has exactly the same caracteristics that the items in the deduction. In fact, it is probably the most obvious problem with continental rationalism and with thought based on it. One remembers this criticism of Hegel by Schopenhauer:

    "Hegel has an incorrect view of the proper relation between empirical reality, which is known by the understanding in perception, and concepts, which are thought by reason in the abstract. In truth, concepts are abstracted from empirical perceptions, and must always refer back to empirical reality to ensure their reliability - or else they can be combined at whim and fictitiously. Hegel, however, erroneously makes abstract concepts primary, and attempts to explain the materialization of the physical, empirical universe from them."

  5. Dr Feser
    A good post, as usual.
    You ask "Have we seen these come together in any recent controversies?" For the purposes of illustration I think you could have backed a better horse than the Covid-19 lockdown controversy. I would have liked instead to hear you apply the principles that you have enunciated to the controversy about anthropogenic global warming.

    1. I assume it would go something like this: should we pretend that we can take the modern world, with its much larger population, back to seventeenth century subsistence farming on the basis of abstract computer models that have never been anywhere close to accurate in projecting the effects of returning a normal element to the atmosphere from which it originally came? We’ll need to kill a few billion people, but, as Michael Scott said “the machine knows!”

  6. It seems to me that there's a conflict for materialists who hold both (a) Scientific Realism with regards to mathematical sciences and models, and (b) Nominalism with regards to abstract mathematical objects (and that's a pretty common pair, especially today).

    You can't have it both ways. If mathematical models/representations gives us even *some* information about extra-mental reality, they (the mathematics) cannot be mere human inventions, and hence nominalism cannot be true. That's why Aristotelian/moderate realism fits with science better. It admits that physics' information is highly abstract, but abstracted from real, extra-mental reality.

    1. BenG,

      Good point. But what can we expect from a movement that uses realism in the proof against realism? A movement that started by denying universals and ends by complaining about "whiteness"?

    2. TN,

      "A movement that started by denying universals and ends by complaining about "whiteness"?"

      That's a trenchant observation. Is there anything more metaphysical and viciously abstract than the "whiteness" that is intrinsic to the worldview of the social justice ideologue?

      When you consider that these folks find "whiteness" at work when younger black girls are harassed on the basis of race by older boys of Indian descent, (, it's clear they're committing a fallacy of reification of epic proportions in spite of purported empirical facts on the ground.

      In the article, Dreher's satirical quip of "the magical cracker" is apt for this noxious absurdity. "Whiteness" or "systemic racism" is apparently responsible for every perceived discrepancy between the races -- from NFL head coach hiring decisions to incarceration rates, the power of the evil "Magical Cracker" is at fault! He's omnipotent and omnipresent, an omnimalevolent demiurge. Like God, he's the metaphysical first principle by which all social reality is subordinated and understood for these people.

      Truly, they're not merely guilty of an error of reasoning, making a fetish of their conceptual categories, or "mistaking the map for the territory," but something that begins to look a lot like idolatry.

    3. BenG,

      I'm definitely not Stardusty, especially due to the fact I'm a conservative Christian as made evident by the comment.

      I apologize if you think I was trolling. I thought TN's remark was especially incisive and interesting. My response was directed toward him and only him. I admit I was a little long-winded and discursive. I didn't necessarily expect a response back.

    4. Hello Stardusty

    5. Modus Pownens,

      I believe you're not StardustyPsyche. Much as we disagree on some items of social justice, he would never be so unthinking as to take the incredibly stupid equivalence Dreher makes as worth consideration. I mean, in a philosophy blog of all places, we should easily see the difference between "They show race in the making, and show how race is something we perform, not just something we are in our blood or in the color of our skin." and "So, even when racist harassers are brown-skinned, they’re really white, and their alleged actions are the fault of white people.", that is, if we want to admit the difference.

    6. ...

      I don't know what I did to garner the attention and comparisons to Stardusty Psyche that I have, but you, One Brow, are clearing looking for a fight. You and whatever happened on this comment subthread are just not worth it.

    7. @BenG

      This a interesting point. The best nominalist answer to why mathematics work in the sciences seems that there is a structure in reality and that humans are slowing discovering it by trial and error. I think that Graham Oppy did defend something like that in his debate with William Lane Craig.

      Now, this answer is pretty much aristoteliam realism, so, yea, nominalists are in troble.

    8. (Man, judging by my errors above i need some sleep...)

    9. Talmid,
      Now, this answer is pretty much aristoteliam realism, so, yea, nominalists are in troble.

      Is Aristotelian realism merely the notion that seemingly identical things behaver in identical ways, and that physical objects influence their surroundings? Because those are the only needed constituents to allow the formulation of mathematical descriptions.

    10. Modus Pownens,

      Some commentators will try to support their positions, some will let disagreements slide, some rare commentators will even acknowledge an error.

      This blog seems to have a high percentage of commentators who prefer to insult and categorize others over discussion.

    11. @Modus Pownens

      Well then I apologise. Your post was indistinguishable from one of Stadusty's. Go and look at how Stardusty responds to posts. It just seemed very overly eccentric and irrelevant to the main subject. But okay you are not him.

    12. @Talmid

      Yep, that's basically conceding Aristotelian Realism. While William Lane Craig did a good job in that debate, I think the incredible "application of mathematics" supports a Divine Conceptualist/Scholastic Realist position rather than a "God as a mathematician" nominalist type of view. God creates the world according the Divine Archetypes - essences, forms, numbers, and so on. It's not at all surprising that mathematics (Euclidian Geometry in particular) has such tremendous application.

    13. @One Brown

      Well, this is not Aristotle view, but it can work out as a more pragmatical view of sciencie. You can do it and your buildings will not collapse. The trouble is when you say that there is a complete order in reality that is described by science, scientific realism, when you try to explain what that order is it is harder to not end up agreeing with the ancient philosopher.


      I agree that it makes more sense that Craig view. His voluntarism on the question "can God make a world with diferent physics" seems bizarre to me.

      And the funny thing is that Dr. Craig does agree with the claim that God creates according to Divine Archetypes, you see he saying it sometimes, it is just that he sees forms as pretty much a God mental creation only. To him, the real work on keeping the world as organized goes, like to Descartes and others, to these laws that God does. Being honest, i agree with Dr. Feser that ocasionalism is the sequel and pantheism pretty much where it ends.

    14. Talmid,

      While I'm sure no offense was intended, this is the second time you added the letter "n" to my handle. I am referring to the hair above my eyes, not their color.

      I agree there is more than reality than described by science, and the some variation of hylomorphism could easily be a part of that. However, it seems to me there are non-Scholastical variations on hylomorphism, and possibly non-Aristotelian as well.

    15. @Talmid

      Worse still, nominalism arguably entails mereological nihilism, something I'm sure Craig wouldn't want to get into the Kalam's 1st premise.

    16. @One Brow

      Sorry about the name :(
      I'am not that good with details, these words are kinda similar...

      Well, there sure is non-scholastic takes on hylemorphism, Aristotle himself would disagree with some of the scholastics, since they themselves disagreed with each other and people like Aquinas did inovate on the greek ideas.

      About the non-aristotelian hylemorphism, that would be pretty interesting to know, actually. Maybe you could argue that people like the platonists, augustinians, cappadocians and others essencialists where close to that, maybe it depends on how we define things.


      Could the nominalist get even mereological nihilism? The mereological simples would need to have particular caracteristics too which would indicate a essence, so it seems that the problem would continue.

      And yea, Dr. Craig is pretty clear on his dialogue with Cosmic Skeptic about find mereological nihilism a bizarre view. Kinda funny.

    17. Talmid,

      As I said, I'm sure no offense was intended.

      Have you read Camels with Hammers? Daniel Fincke was trying to create a version of hylomorphism somewhat different from the Aristotelian version. I don't know if he still is.

    18. @Talmid,

      Wait, did Oppy actually say that there is most likely real structure in reality to explain why math is so good at modelling the world?

      Did that really happen?

    19. @One Brow

      I had never heard of it before, so thanks for the tip. I read a post from the guy(i think) about hylemorphism and i can see how you could use the concept in a non-aristotelian context.

      But i don't know if Daniel sees himself as non-aristotelian. From his post, he does not seems to find a problem between his more computational view of the idea and Aristotle himself. I don't know if the greek actually believed in life after death and all that, that seems pretty controversial,so maybe they are not THAT diferent.


      Its been a while since i saw the debate, so i could be exaggerating. But i remember he saying something like that yes, thought i doubt that he is a realist on universals. Oppy seemed to me to see our mathematics as derived from reality.

      I even remember that when asked by Dr. Craig on why we have these laws and not others he replied that they are necessary*, in the sense that it could not be others(which i agree). But again, its been a while, so watching the debate would be more trustworth.

      *a strange thing is that i remember Oppy wanting to finish the whole question of the origin of the laws there, like if he thought that a thing being necessary means that it can't have any explanation, but i could be remembering wrong

  7. The never-dying, misguided belief that we can understand and control complex phenomena--the fatal conceit.

    The moderns wanted to replace religion/metaphysics with the certainty of science. Now they just put the term “science” on their religion. Math can’t be a religion, but racist math can be; all we need is a different priestly caste.

    There are no day traders on the Forbes 400, but scientists (and Harold Camping) are freed from the burden of having their account drained by a wrong bet.

  8. The way scientists and others so often treat the abstraction as the thing is deeply entwined in scientism. This is why we have such problems in quantum mechanics where the wave function is like a translation between the form and it’s representation when observed/measured (such that the form expresses itself as matter). Because what is being studied is by definition metaphysical, the usual method of ignoring the metaphysical distinction breaks down.

    On the Covid point, it’s worth adding the precautionary principle. These things can get out of control very quickly. In this case we’re lucky that it has a relatively low IFR, is not a big threat to kids etc, but if it was something more like ebola there would be no going backwards from erring on the side of “wait and see”.

  9. @JoeD

    Where would you throw the nonmeasurable target?

    Prove that your "probability bounds" are at least greater than 0 or less than 1.

  10. Wolfgang Smith is about to be taken off the back burner. But first I'm going through Christopher Norris's Quantum Theory and the Flight from Realism line by line. Reads like a Novel. Smith is also an engaging writer. Both are beasts and anything they write is worth reading. I'd like to see Ed's thoughts on their works.

    1. You do have a blog, I would also like your thoughts on him as well. If I remember correctly you used to be an atheist and have since changed your mind.

    2. Hey Tap,

      Correct. My book, out by year's end (Chinese, Russian, & English, title under wraps), will be a comprehensive refutation of scientism, anti-realism, and atheism in about 140 pages. One entire page is probably going to be Ed's entry in the bibliography.

  11. The dogmatic scientism that motivated this disaster in policy

    I would point out a modest refinement regarding this comment by Ed: It is probably not really "scientism" that is the culprit here. I suspect that's the wrong word.

    Here's what I mean: Let us suppose that we take "scientism" to mean mean, say, either (1) the erroneous conflation of the abstracted model (and rules of manipulation) with the concrete reality, (i.e. the major point of this article), or (2) the related erroneous assumption that the physical world AS CONCEPTUALIZED without formal or final causality constitutes the whole of reality that can be studied for knowledge, then either (or both) of these are primarily errors by scientists about the relationships between abstractions and reality.

    The difficulty is that both the COVID stuff and the (alternatively suggested) global warming political nightmare stuff are primarily erroneous applications of what are, in reality, extremely preliminary and extremely tentative initial findings about facts where, eventually, science may really be able to arrive at reliable and mature conclusions regardless of any erroneous metaphysical commitments. And while these political nightmare behaviors MAY be supported by (politically motivated) scientists who are given to scientism, they are primarily being made by politicians who haven't enough science to be ABLE to make the erroneous scientific metaphysical assumptions implied in "scientism" themselves. And further, the politicians would be (mis)-using the extremely preliminary and tentative scientific findings REGARDLESS of whether or not they had any pet scientism-based scientists at hand to assist them.

    Granted, there may be an unholy marriage or synergy (or, "sinurgy") between the scientists and the politicians who provide the grants for the scientific endeavors, but at least with COVID, the sinurgy lies primarily in the awful pride on the part of both about "what we know" and "what we can do about it". Perhaps scientism gives scientists a greater bent toward such pride, but it doesn't give them a reasoned basis for thinking that extremely preliminary findings are identical to mature scientific conclusions, and THERE ARE OTHER SCIENTISTS who have been saying "whoa, nellie, you guys are jumping way too fast on way too incomplete an analysis." See, for example, The Great Barrington Declaration.

    I had hoped that this epidemic would provide the world with a great test-case for figuring out what to do in case of a REALLY BAD disease, one with, say, a 10% IFR instead of the roughly 0.3% IFR we (perhaps) have here (i.e. 30 times worse than COVID). Or with one that has a MUCH longer incubation / re-transmission period before symptoms than COVID has. Or both, naturally. Unfortunately, now that the sheer science has gotten totally politicized, there is no strong prospect that any solid, dependable science will be gleaned from this until at least a decade, but more likely 2 decades from now (i.e. after the careers of all senior people involved are moot).

    1. @Tony:
      I agree, "scientism" is not the culprit. I continue to be baffled as to why Feser thinks it is relevant. Scientism is, I take it, a species of Bad Philosophy of Science, and BPoS is arguably relevant (although the problem mostly seems to be Bad Science, i.e., BS rather than BPoS), but not scientism in particular.

      As for what scientism is, I wouldn't have said it was either of the options you mention. I thought it was the straightforwardly naive (usually completely unanalyzed) claim that the only real knowledge is scientific knowledge.

      Anyway, certainly a good scientist can be a bad philosopher of science (or bad at public policy), just as a good person can be a poor moral philosopher.

      As for figuring out what to do in case of a Really Bad Disease, I wonder if that's a bit like getting hit by a small volcanic eruption, and hoping that now we'll know better how to handle a 30 times bigger eruption. Maybe... not? Or maybe it's like starting a small fire on an ant hill and wondering, as you watch the ants freak out, if this will help them figure out how to deal with a 30 times bigger fire in the future. (Obviously ants aren't even potentially rational so it's an imperfect analogy, but...)

    2. With the volcano, I'm thinking of the good old legal concept of an Act of God, i.e., there are things that are "outside human control ... for which no person can be held responsible" (says Wikipedia) and which are beyond our power to "figure out" -- worth pondering, perhaps.

  12. You hoped the epidemic would provide the world with a great test-case of figuring out what to do in case of a REALLY BAD disease did you ? Lives are clearly of no consequence to you then Tony. Sick.

    1. @ Anonymous. I did nor hear Tony say that.
      @ Tony Unfortunately you cannot separate out science and metaphysics in the way you seem to hope. What certainties we have gleaned from science are based on asumptions that are metaphysical, not scientific: such as the regularity of natural laws and the validity of reason.

    2. Anon: I was hoping that we would be able to learn things from THIS disease (with a 0.3% IFR) that would stand us in good stead and enable us to learn how to react to a much worse disease like one with an IFR of 10%. Like most others (including scientists), I hope that in planning against disasters, we can employ or design "test cases" which INFORM us without SUBJECTING us to dinosaur-killer level disasters. I do trust you to agree with me that a disease which has similar characteristics to COVID except for having an IFR of 10% would be "much worse".

      Jonathan, I am OK with the fact that science goes forward with a set of metaphysical underpinnings, which are not themselves wholly subject to the scientific method for verification. My main point is that the foolish governmental behaviors we have seen are not, per se, behaviors of scientism, but of some other disordered take on things. A better word than "scientism" should be found for it. There are scientists who are (more or less) accepting of some of scientism's flaws who STILL think that governments shouldn't have acted this way. And when, in the course of 20 years of further analysis of this past year's unfolding, we arrive at mature conclusions by scientists who are entirely in the camp of scientism, it is extremely likely that one thing that comes clear is that politicians used preliminary, tentative findings from scientism camps as if they were mature, developed, SETTLED results from scientism camps, and for that very reason they were wrong-headed even in the cases where they lucked out by accidentally landing on a successful(ish) moment like a broken clock. In such case, it isn't the scientism of the findings but the USE of them that is main problem. (I have no problem using an airplane designed by physicists and engineers who are given over to scientism - as long as they are using mature, settled science for their design work, not slapdash preliminary work.)

  13. Dr. Feser - have you looked into the formal ontology BFO of Barry Smith (Monist editor)? It is now an ISO standard based in Descriptive Logic, is the basis for the Gene Ontology (Human Genome Project), and has just been adopted by the Department of Defense.

    It seems to me that a BFO physics ontology carefully informed by hylomorphism would be a major contribution (for computational purposes, practical applications, and philosophy)- and BFO still lacks a standard ontology for physics.

    Would IMO be great to have Maudlin, Albert, Sheldon Goldstein, Oderberg, maybe Eric Weinstein on the project rather than leaving the basic conceptual framework to those who would want "quantum measurement" and "quantum observer" as basic categories.

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