## Wednesday, January 31, 2018

### David Foster Wallace on abstraction

In his book Everything and More: A Compact History of Infinity (he had a way with titles), David Foster Wallace has some wise things to say about abstraction.  To orient ourselves, let’s start with the definition of “abstract” he quotes from the O.E.D.: “Withdrawn or separated from matter, from material embodiment, from practice, or from particular examples.  Opposed to concrete.”  So, for example, a billiard rack or a dinner bell is a concrete, particular material object.  Triangularity, by contrast, is a general pattern we mentally abstract or separate out from such objects and consider apart from their individualizing material features (being made of wood or steel, being brown or silver, weighing a certain amount, and so on).

Mathematics is a paradigmatic example of a discipline that deals with abstractions, and it is the one that is the focus of Wallace’s book (though he has some things to say about philosophy too).  He notes how children are introduced to numbers:

First they are given, say, five oranges.  Something they can touch or hold.  Are asked to count them.  Then they are given a picture of five oranges.  Then a picture that combines the five oranges with the numeral ‘5’ so they associate the two.  Then a picture of just the numeral ‘5’ with the oranges removed.  The children are then engaged in verbal exercises in which they start talking about the integer 5 per se, as an object in itself, apart from five oranges.  In other words they are systematically fooled, or awakened, into treating numbers as things instead of as symbols for things.  Then they can be taught arithmetic, which comprises elementary relations between numbers. (pp. 8-9)

This little narrative nicely illustrates the sense in which an Aristotelian would regard numbers as abstract objects.  (To be sure, Wallace himself was not an Aristotelian, but it turns out that some of his examples and remarks are readily adaptable to Aristotelian purposes.)  Numbers are objects in the sense that we can refer to them, attribute features to them, etc.  They are abstract objects in the sense that they exist only as abstracted out from concrete reality by the intellect.  There can be five apples in mind-independent reality, or five dogs, or five donuts, but not five itself, considered as a kind of freestanding entity.  The number five is not an “abstract object” in the Platonic sense of the term common in contemporary philosophy (viz. an object that exists in some third kind of way over and above intellects on the one hand and concrete particulars on the other), and could not be.  For what is abstract is, for the Aristotelian, what results from abstraction, and abstraction is a mental activity of considering one aspect of a thing in isolation from other aspects.

The example also indicates, at least obliquely, both the power and the perils of abstraction.  That mathematical reasoning is powerful goes without saying, and it would not be powerful if it didn’t get at something deep in the nature of objective reality.  The intellect doesn’t make up mathematical truth out of whole cloth, but pulls it out from the concrete reality into which it is, as it were, mixed.  The peril is that we can easily be “fooled” (as Wallace puts it) into treating mathematical objects as if they existed outside the mind in precisely the abstract way in which they exist within the mind.

Nor is it merely that abstract objects – numbers, geometrical figures, universals, etc. – do not exist in this third, Platonic way.  Another peril is that we are tempted to take what are really just artifacts of the abstractive exercise to be features of mind-independent reality.  Commenting on Zeno’s dichotomy paradox, Wallace writes:

[T]here’s obviously some semantic shiftiness going on here… [which] lies in the implied correspondence between an abstract mathematical entity – here an infinite geometric series – and actual physical space… [T]raversing an infinite number of dimensionless mathematical points is not obviously paradoxical in the way that traversing an infinite number of physical-space points is… [T]he translation of an essentially mathematical situation into natural language somehow lulls us into forgetting that regular words can have vastly different senses and referents.  (p. 70)

As the example of Zeno shows, the tendency to confuse abstractions with concrete reality is the source of many metaphysical errors.  To be sure, as Wallace immediately goes on to say of the specific muddle he calls our attention to:

Note… that this is exactly what the abstract symbolism and schemata of pure math are designed to avoid, and why technical math definitions are often so numbingly dense and complex.  You want no room for ambiguity or equivocation.  Mathematics… is an enterprise consecrated to the ideal of precision.

Which all sounds very nice, except it turns out that there is also immense ambiguity – formal, logical, metaphysical – in many of the basic terms and concepts of math itself.  In fact the more fundamental the math concept, the more difficult it usually is to define.  This is itself a characteristic of formal systems.  Most of math’s definitions are built up out of other definitions; it’s the really root stuff that has to be defined from scratch.  Hopefully… that scratch will have something to do with the world we all really live in. (pp. 70-1)

Wallace seems to think that the way people study math at more advanced levels tends to exacerbate the problem:

The trouble with college math classeswhich classes consist almost entirely in the rhythmic ingestion and regurgitation of abstract information, and are paced in such a way as to maximize this reciprocal data-flow – is that their sheer surface-level difficulty can fool us into thinking we really know something when all we really ‘know’ is abstract formulas and rules for their deployment.  Rarely do math classes ever tell us whether a certain formula is truly significant, or why, or where it came from, or what was at stake.  There’s clearly a difference between being able to use a formula correctly and really knowing how to solve a problem, knowing why a problem is an actual mathematical problem and not just an exercise. (p. 52)

Now, this is an extremely important point, which applies well beyond mathematics itself.  The sheer difficulty of reasoning about abstractions can lead us to overestimate the significance of the payoff, especially when the payoff is indeed significant in some respects.  Nowhere is this truer than in modern physics.  As Wallace writes:

The modern transition from geometric to algebraic reasoning was itself a symptom of a larger shift.  By 1600, entities like zero, negative integers, and irrationals are used routinely.  Now start adding in the subsequent decades’ introductions of complex numbers, Napierian logarithms, higher-degree polynomials and literal coefficients in algebra – plus of course eventually the 1st and 2nd derivative and the integral – and it’s clear that as of some pre-Enlightenment date math has gotten so remote from any sort of real-world observation that we and Saussure can say verily it is now, as a system of symbols, “independent of the objects designated,” i.e. that math is now concerned much more with the logical relations between abstract concepts than with any particular correspondence between those concepts and physical reality.  The point: It's in the seventeenth century that math becomes primarily a system of abstractions from other abstractions instead of from the world.

Which makes the second big change seem paradoxical: math’s new hyperabstractness turns out to work incredibly well in real-world applications.  In science, engineering, physics, etc. (pp. 106-7)

But this tremendous effectiveness can be misleading (and I should note that in what follows I go beyond anything Wallace himself says).  Modern physics is very difficult indeed; unlike mathematics, it is concerned with the physical world; and the payoff in predictive and technological success is enormous.  The temptation is strong to conclude that everything in the mathematical model of the world presented by physics corresponds to something in physical reality, and that there is nothing in physical reality that isn’t captured by the model presented by physics.

But it simply isn’t so, and the mathematical abstractness of physics is precisely what guarantees that it isn’t so.  Abstraction by its very nature leaves out much that is in concrete reality, and the more abstract the model arrived at (as when, to use Wallace’s nice phrase, we are dealing with “a system of abstractions from other abstractions”), the more that is left out.  Physics can no more tell you everything there is to know about the material world than learning how to count oranges can tell a child everything there is to know about oranges.  As Bertrand Russell put it in a passage I’ve often quoted:

It is not always realised how exceedingly abstract is the information that theoretical physics has to give.  It lays down certain fundamental equations which enable it to deal with the logical structure of events, while leaving it completely unknown what is the intrinsic character of the events that have the structure… All that physics gives us is certain equations giving abstract properties of their changes.  But as to what it is that changes, and what it changes from and to – as to this, physics is silent. (My Philosophical Development, p. 13)

Or, as Russell put it more pithily and wittily elsewhere:

Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract.  Only mathematics and mathematical logic can say as little as the physicist means to say.  (The Scientific Outlook, p. 82)

Like Zeno, contemporary popularizers of science, and sometimes scientists themselves, confuse mathematical abstractions with concrete physical reality and draw absurd metaphysical conclusions.  This is precisely what happens when it is claimed that relativity has shown that time and change are illusory, as Lee Smolin and Raymond Tallis have recently pointed out.  Indeed, as Tallis emphasizes, the tendency in modern physics is to abstract from the notion of time whatever isn’t space-like, but also to abstract from the notion of space everything but pure quantity – and thus to treat time as an abstraction from an abstraction, in other words.

But again, now I’m going beyond Wallace, so I’ll stop.  Much more on these subjects forthcoming in the philosophy of nature/science book I am currently working on.

1. Dr. Feser,
I recently read your Article about Act/potency and block world. Whats interesting and relevant here is your remarks about "Ontic Structural Realism" Do you consider the particular objections you make to be decisive?
Always looking for more remarks on Ladyman and Ross's book as I find their book one of the most rich and sophisticated defense of Naturalism. OSR seems very nicely motivated and main and a viable rival to A-T metaphysics and view you express above.

1. Hi Red, I do think so, but of course the remarks in the paper are only brief. I have a lot more to say about Ladyman and Ross in the forthcoming book.

2. How did you get hold of his paper, Red? The Google books preview doesn't provide the whole thing.

3. Can you link to act/potency article?

5. Im sure you have, but in working on your forthcoming book did you engage/read Tahko. I think he has some interesting things to say to Ladyman and Ross.

2. Ed, If I remember correctly, is this philosophy of nature book due to be published around May? For some reason I seem to recall you giving an estimation for that due date last year. I have more than a sneaky suspicion that I have pulled that from my behind though.

3. Thank you for this very interesting article. I'm a mathematician, and think about these kinds of things a lot, though I haven't made up my mind quite what I think about it all. (As I'm sure you know, mathematicians "feel" the objects they deal with as having reality and character that is independent of their mind.) Anyway, I'm going to think over your post and come back if I have anything relevant to say, but for now, thank you for dealing with this difficult area.

1. Perhaps this is what you meant by "reality [...] independent of their mind", but for clarity's sake, the point is that things like numbers and Triangularity do have a reality outside the mind, but not as independently existing things. There's no Triangularity in the world as an object per se, but that doesn't mean that Triangularity isn't exhibited by things in the world. (I have heard that many mathematicians are, tacitly or consciously, Platonists such that they do hold that things like numbers exist in their own right, and perhaps that's what you meant in your post.)

I'm not sure how some of the other mathematical objects, like numbers, exist in the world though. Would like to hear Dr. Feser's views on this topic.

2. Yes, you're correct, I meant it in the Platonist, externally existing sense. I appreciate your clarifying post. I tend not to express myself very well in philosophy.

One might consider, for example, the "space of all square integrable functions on a sphere." This is a very abstract thing, and it seems a stretch to say it's abstracted from the world per se, but at the same time, it's a geometric space (infinite dimensional) with its own properties and "character," and to one who works with it, it can seem awfully real (and independent). That's not conclusive, but I do think the very common reaction of workers in a field toward what they're working on is at the very least interesting. (Nor is this analogous to scientism, which many physicists adopt. Scientism is in some sense the reigning orthodoxy in the academy. On the other hand, Platonism is regarded askance by many if not most philosophers of mathematics, but mathematicians often arrive at it anyway.)

3. What about computer models? They are surely objects of some kind. In computer animation, complex geometry is used to make graphic object that look real. When we see objects created by CGI, they look like real objects usually. But then maybe this isn't so different from an image drawn on graph paper.

4. I'm not sure I follow your point's application, Jonathan. Would you mind elaborating?

5. works of art are often created using the laws of perspective. a painting of a road will show the lines of that road converging on a point in the distance and the outlines of buildings will be at an angle that extends toward the same vanishing point in the distance. Even though this space is illusory, the real world is thought to obey the same laws of perspective. But people talk about the laws as if these laws were something separate from both the painting and the physical world. When we say the world "obeys" the laws of logic or nature or perspective, it sounds like those laws are something distinct from the world itself. This might be what leads people to a Platonic philosophy.

4. I'm confused about the commentary on Zeno.

"[T]raversing an infinite number of dimensionless mathematical points is not obviously paradoxical in the way that traversing an infinite number of physical-space points is…"

But wasn't that Zeno's point? He was saying that it is impossible to move at all because to traverse an infinite number of physical-space points, which exists between any two physical-space points, is impossible.

Is it that there can be infinite mathematical points between the two points, but not actually an infinite number of physical points?

1. Maybe because a point has no magnitude. So it's not occupying space.

2. Because it doesn't have any magnitude is precisely why there is an infinite number of points between two points.

4. Zeno's paradox of the arrow might be a better example of the impossibility of motion because it divides time instead of space. When an arrow is shot from a bow, at any one moment of time it occupies a certain amount of space. Considered at that moment, the arrow is not moving because movement is a changing of location. The arrow can only be where it is. It can't be where it isn't. So how can it be moving from one place to another?

5. Dr. Feser,

What do you think about polish Catholic priest and physicist Michal Heller's objections to A-T metaphysics? His criticisms touch on the subject on how seriously we should take abstractions.

For example, he criticises Jacques Maritain's complaint that mechanistic mathematics deals only in quantity and naturally by it's very method excludes any qualitative aspects, to which Heller responds that mathematics is "so much more" than mere quantity, but about logical structure and making rock-solid deductions about the world that cannot be mistaken.

He used to study Thomism during the seminary days of the 50's, but rejected it afterwards after being amazed by the marvels of modern mechanistic science.

What do you think?

1. math seems unable to describe the real world as it is because the math is just quantities, but the actual world cannot be made of quantities alone because they wouldn't be quantities "of" anything.

2. Joe, I would suggest that anyone who abandon's Thomism because of amazement at the modern mechanistic science is making a grave category mistake. And clearly didn't get Thomism. And almost certainly should not be a Catholic priest. There is nothing in modern science that is in opposition to Thomism except the bad philosophy of physicists who think that they can do philosophy because they can do physics. Lawrence Krauss and Stephen Hawking give good examples of bad philosophy done by physicists.

3. @Tony,

Here are a few things about Michal Heller's comments on Thomism from the Polish wikia that should prove interesting:

" He emphasizes the merit of Thomism in the formation of modern science , but in his opinion it is an outdated, obsolete philosophy, which is one of the reasons for tensions between faith and science in modern times .

Through this criticism of contemporary Thomism he has called himself a "Christian positivist", deprecating the role of scholastic metaphysics and the clinging to concepts like matter and form , which in his opinion are for the physicist to name the same in a different way"

6. Gack. Wallace was such a degenerate. All the muck about him owning copies of illegal porn in his books is disgusting. We really should stop giving people a pass on this stuff just because "he's, like, an artist man, you just don't understand".

His writing was awful too. Naval-gazing nonsense.

On abstraction and mathematics, try Berkeley's The Analyst. It's witty. Unlike Wallace's turgid, obnoxious, know-it-all geek-prose.

https://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.pdf

1. Thanks for the Berkeley essay but where on Earth did you hear that DFW had "illegal porn" in his books?

2. I've read a number of Wallace's essays and listened to him give the commencement address on water. From this I gather his was a great, intelligent, beautiful, and troubled soul.

I'm really sorry if you lost a loved one to suicide, but there's no need for character assassination on someone else who committed suicide.

The stuff that was said about Robin Williams after his death was horrifying. No need to repeat that calumny with Wallace.

7. Very interesting. I did not know that David Foster Wallace is writing on such topics.

The following spins off from Prof. Feser's OP, about abstraction of mathematical entities (?) in Aristotle.

"Any so-called square or circle which we see is not a genuine square or circle, but only an approximation to one; yet geometry does discover properties of the perfect square and of the perfect circle. Reason has to *divine* the existence of perfect squareness and of perfect circularity; and Aristotle expresses this with a certain measure of exaggeration, by saying that reason has to make them, to think them into existence, and in doing so is purely active, and not dependent in any degree on sense-perception of perfect squares or perfect circles..." W.D. Ross, Comm. on Aristotle's De Anima, p. 46.

1. We are accustomed to speak of "the active intellect." But should we add some notion like "constructive," with the idea that Ï€Î¿Î¹Î·Ï„Î¹Îºá½¹Ï‚ contains the idea, "able to make"? Aristotle doesn't use that adjective directly, but he implies it when he speaks of the passive intellect as Ï€Î±Î¸Î·Ï„Î¹Îºá½¹Ï‚ and when he says things like "intellect, Î½Î¿á¿¦Ï‚, makes, Ï€Î¿Î¹Îµá¿–, all things" (DA III.5, 430a15). The thought that the activity of abstraction is a kind of "making" of perfect triangles or whatever, which qua perfect are not found in perceptible bodies, seems to fit.

2. What kind of entity is a mathematical abstraction in Aristotle? Not all mathematical entities are properties of identifiable material substances/perceptible bodies - witness some of Foster Wallace's examples. But they aren't Platonic Forms. So what kind of entity are they for the Aristotelian? Esp. as we get away from geometry into other branches of math.

8. Dr. Feser, in what manner does, say, the number 5 exist?

If the intellect abstracts Triangularity from individual triangles, Human from particular human beings, Chairness from individual chairs, etc., what exactly is the subject from which numbers are abstracted? In other words, 5 oranges are 5 independent substances taken together with nothing unifying them except my possibly arbitrary decision to entertain them as a group.

1. In her commentary on Metaphysics Books M and N, Julia Annas emphasizes that for Aristotle, number is always related to objects that are being counted, and the unit is the measure of the number. Units are to counting as, say, inches are to measuring; the unit of counting and the unit of measuring are both choices of measure. "... the unit in arithmetic is what is taken as indivisible for the purposes of counting or computation. Thus the problem of the indivisibility of the mathematicians' unit is solved without Plato's postulation of perfect pure and indivisible units. The mathematician's unit is just an ordinary physical object *regarded as* indivisible for counting; it is not a different sort of object altogether... Aristotle's concept of number, in making number relative to what is numbered, ties number firmly to counting, making it analytic that number is what we count with. This is a suitably anti-platonist theory: numbers do not exist independently of us and our activities of counting. (Cf. Physics 223a16-29, where Aristotle concludes that there would be no time--which for him is a kind of number--if there were no conscious beings to count.)" (pp. 37, 39)

Annas criticizes this theory as too restricted, e.g. it does not allow fractions to be numbers, since we might be able to count by halves but "we can hardly count with 17/102".

2. As I understand it, that doesn't seem to really address the claim that mathematical objects are abstracted from reality, or how. It sounds to me like you're begging the question by assuming counting. But how can you count without already knowing number? Let's say I take something to be the unit. The only way I could take something as unit is if I already understand unit which is itself numerical. Then, if I were to talk about 5 units, I have to invoke the number 5 as well. But where does that number come from? If it comes from unit, then we have to understand not just unit (a number), but also multiplication.

In line with the one and the many, common sense tells me there is a multiplicity of things in the world, a plurality of unities. But again, that already presumes the understanding of number, as does measure, as does division.

3. @Anonymous of Feb. 1,
It is not I who is begging the question, if the question is begged, but Julia Annas, whom I simply quoted (heh heh).

I don't see question begging, though. Isn't it one thing to ask, what kind of entity is a number, or where does it come from, and another thing to ask, how do we learn to count/manipulate numbers? Surely you're not suggesting that people who count have not already known numbers.

Annas is trying to figure out Aristotle's criticism of Plato's mathematical entities and Form-numbers (as well as his criticisms of Speusippus and Xenocrates).

9. It seems like your description of math applies instead to physics. In physics parts of the world, admittedly very limited parts, are described and pulled out. However in math it seems that although parts are pulled out, at least some of the essence is left in the mathematical object. For example mathematics displays what I call lateral utility instead of just utility. Lateral utility is utility in completely different things. Math displays this. However the essence of an object merely as a part, say the 5 in 5 oranges could never have reference to the 5 in 5 boats and if it could it would be simple utility. But with lateral utility the essence of the orange could clearly have something to do with the essence of the boat.
Whereas normal utility would be described as a part having increased utility in virtue of it being a part of something
else. A classic example would be mass being a part of everything.

All that to say math does not really describes parts.

10. Zeno was right. if you take a ball and divide it in half, and then again, to infinity, you have an infinity of parts. Aristotle was wrong to say its only potentially infinite. If it can be divided, it has those parts

1. Huh? You can divide a ball in half an infinite number of ways (at least geometrically speaking). Which halves are the parts of the ball? Clearly, not all of them since that would mean you no longer have two halves.

2. Except you can't actually count to infinity or divide your way to it. At best you can abstract your way to it without counting. The Ball can be potentially divided but it doesn't actually have those parts till you actually divide it.

Don't believe me?

Try this experiment. Divide your way to infinity (or count if it's easier) when you are done come back and tell me & I will concede you are right here.;-)

3. Actually, you are both wrong, for two reasons.

1) If you take a ball, and divide it up infinity and place the biggest half on the left and so on to infinity, on the right side the series will go on forever

2) Just because a stick is not divided into two pieces doesn't mean the pieces divided don't equal the size when all put together.

Kant was right. To pass over a bridge you have to cross over all the planks, and this process goes on forever. Nothing stops it. So all the points must be crossed. This is called on "antinomy". Something can't be both infinite yet finite at the same time. Yet everything is. Have you heard of the Banach-Tarki paradox?

4. Actually, you are both wrong, for two reasons.

You've missed their point entirely; you have never divided a ball infinitely and never will, and there are no actual infinitesimal parts that you can identify without assuming a process of division that has gone on forever. If you cross a bridge, you aren't crossing points; you are crossing planks. You don't cross infinite things, one magnitude-less point after another; you crossed finite things which you can imaginatively divide and keep dividing, the divisions creating points. You can't get the antinomy unless you assume that you've already divided the line infinitely many times; and as Son of Ya'Kov noted, you are a liar if you say you have.

5. So a yard of wood doesn't really have three feet? Your fingers can feel each foot. The process goes on for ever. A ball divided infinitely has the same mass when it is put together

6. So a yard of wood doesn't really have three feet?

If you are dividing a yard into feet, you have divided it into a finite number of feet. That is all.

The process goes on for ever.

This is precisely the point. Which process? You can't point to any. You have never divided a yard of wood into infinitesimal points. You can't point to any natural process that divides a yard of wood into infinitesimal points. It's not, in fact, even possible to divide wood into infinitesimal points; wood, as such, is not infinitely divisible.

The problem, as noted before, is that you are assuming that the points are given. No Aristotelian does; points are seen as termini, and therefore it makes no sense to say that there is actual a terminus if you've never established anything for it to terminate. Thus you are begging the question, and assuming that if you could construct a point that the point is actually constructed.

A ball divided infinitely has the same mass when it is put together

You have never divided a ball infinitely, nor can you point to anyone or anything that has, or any process capable of doing so. What you are doing, from an Aristotelian perspective, is confusing the abstract intelligible structure of dividing-a-sphere constructions with actual objects of spherical form.

7. My response to Anon.

What Brandon said....nice job bro! Thanks for the assist.

8. Aristotle was wrong about this. Your position is that you can cross a bridge without crossing the planks, a yard without crossing the feet, a foot without crossing 6 inches. I can do this forever. The points are there because we are talking about matter. If a banana is divisible in two, then it has those two parts. If it is infinitely divisible, then it has those parts. This is all very obvious. See Vsauce's video on the Banach-Tarski paradox on youtube. It might help you

9. Your warmed over Zeno/Parmenides nonsense is refuted by the fact I can in fact cross bridges, planks, a yard, a foot, and 6 inches. Ask Diogenes the Cynic.

>I can do this forever.

A banana is merely potentially infinitely divisible and not actually so. Don't believe me? Go and actually infinitely divide it then come back and I will concede defeat.

Good luck with that.

PS. What is obvious is your whole scheme falls apart once you accept the distinction between potential vs actual.

10. Your position is that you can cross a bridge without crossing the planks, a yard without crossing the feet, a foot without crossing 6 inches. I can do this forever.

Both of these claims are obviously wrong. When you cross a bridge you cross what can be divided into planks; when you cross a yard, you cross what can be divided into feet; when you cross a foot, you cross what can be divided into inches; and so forth. You do the division in the measuring. And it is a blatant falsehood that you can do the division forever; and, what is more, no matter how long you do it, you will only have divided the line finitely. You are, again, making the mistake of treating the structure of dividing-a-line (that one can always do it again) is the actual structure of the line (that it is already actually done). What we can say is that for any operation of division on the line another, finer-grained division can be done. But this is precisely what Aristotelians mean by saying that the infinity here is potential rather than actual, and that the line is infinitely divisible, not infinitely divided.

The points are there because we are talking about matter. If a banana is divisible in two, then it has those two parts. If it is infinitely divisible, then it has those parts.

No material things are infinitely divisible. If you divide a banana, you reach a point at which your 'parts' are no longer banana; a banana as such, like wood, can't be divided further than its cells, bodies can't be divided further than atoms, and by the time we get to atoms we are in a realm in which it is literally hazy what we are even dividing in the first place. Only purely intelligible structures -- mathematical lines, planes, volumes, etc. -- admit of division without restriction.

11. @Brandon,

Only purely intelligible structures -- mathematical lines, planes, volumes, etc. -- admit of division without restriction.

Would this also apply to the Banach-Tarski paradox? I remember reading how some theoretical physicists conjectured how some quantum mechanical events can be explained by appealing to the existence of things that are infinitely complex and are being rearranged as to make new copies of themselves.

I also have one more important question.

Considering that theoretical physics deals with heavy abstractions and modelling that might end up having nothing to do with concrete reality as it actually is (Cf. Bas Van Fraassen, The Scientific Image ), what do we make of instances where highly abstract mathematical theorising actually ends up correctly describing reality, such as the discovery of the existence of the Higgs Boson and confirmation of basically every single property it was theorised to have, all of which prior to 2012 were considered to be in the realm of unproven speculation?

12. It is probably useful to add that if one is doing physics like classical mechanics or GR one needs differential geometry. But to do differential geometry it is *not* necessary to assume geometric figures (e.g. manifolds) are made of points; there are other models like (well-adapted) smooth topoi, and in these you *cannot* pull up stunts like the Banach-Tarski paradox.

13. There is no such thing as a potential infinity of matter. To cross over a finite object is to cross over its parts. Those parts are finite, but have parts. Its as clear as the noon day sun that those parts have parts as well. To say there is a part that cannot be divided is demonstrably false. Take the smallest finite part and make another of it. Draw a hypotenuse connecting them and cut the hypotenuse in half. There is a smaller part! You are ignoring the fact that every "finite" thing has parts, which must also have parts, and so on to infinity. You have to think harder on this. Just because a banana ceases to be a banana at a certain point doesn't mean that it cease to be matter and ceases to have parts which have parts which have parts. It is part of the PARADOX that a finite thing is infinite and motion takes an infinity of steps. Do you deny that when you try to move, you must move half of that space first? Where does it end?

14. JoeD,

The primary point I was making was that the inference to existing (and presumably material or material-like) points on the grounds that we were talking about material objects to begin with was an illicit one. You can't divide banana infinitely; you eventually start getting nonbanana and, indeed, beyond a certain point it gets very difficult to figure out what you are dividing and how. The mathematics of material objects, though, doesn't have this problem. There could be all sorts of infinites involved in material objects this way; that's just a question of what is most appropriate for explaining their behavior.

But with infinite divisibility in particular, what one is always talking about is a thing's geometrical properties with respect to operations that could be done, unless you identify the actual operation. You can get a perfectly legitimate kind of infinite divisibility this way (it's an error, and misconceives the whole point of how potentiality works in Aristotelian systems, to treat potential infinity as not a legitimate kind of infinite); it's just, as Aristotelians have always said, a potential infinite.

what do we make of instances where highly abstract mathematical theorising...

There's a tendency to think that modeling is somehow a purely conjectural exercise, but this is the opposite of true; you are modeling real things, and if your experimental and observational evidence is rich enough, and your causal inferences and formal inferences correct, one would expect a model to be quite good, because it's actually capturing what's really there, to whatever approximation and precision one's modeling assumptions allow. As William Wallace has always pointed out, the Aristotelian position has always been that when we talk about things being approximately true, the 'approxiately' is not an alienating adverb -- approximately true is not (as positivists and Duhemians tend to treat it) 'not quite true' but true, to the degree the approximation allows.

15. To say there is a part that cannot be divided is demonstrably false.

Nobody is saying that there is a part that cannot be divided. The point is that the assumption that 'can be divided' and 'divided' are the same is also blatantly false.

Do you deny that when you try to move, you must move half of that space first?

When you move -- not 'try to move' -- you cover a space that can be divided into halves. That is all.

16. Matter is the physical manifestation of geometry, so what apples to the latter applies to the former. How can you deny that something has parts, which has parts, which has parts, forever? How can something have parts potentially? That is what you are maintaining

17. >Matter is the physical manifestation of geometry, so what apples to the latter applies to the former.

Why should we believe that?

> How can you deny that something has parts, which has parts, which has parts, forever?

How can you logically know it has parts forever? You would have to examine all it's infinite parts which is the same as counting to infinity.

My challenge remains open.

> How can something have parts potentially? That is hat you are maintaining.

Are we being punked? Is this Star Dusty?

18. Matter is the physical manifestation of geometry, so what apples to the latter applies to the former.

And, again, the Aristotelian does not assume that geometrical infinites are actual as opposed to potential; and, again, as I have previously pointed out to you, the Aristotelian would say that the structure you keep attributing to lines, etc., is actually the structure of division as such, and only of lines insofar as they can be divided by particular actual operations of division. That 'can be' is precisely what is meant by 'potential'. This has already been pointed out more than once.

How can you deny that something has parts, which has parts, which has parts, forever?

Again, you have not established that this is even possible; the Aristotelian does not assume that it is unless you can identify the specific construction that makes it so. You have not done anything like it, nor in the case of infinite divisibility can you do so: you have no actual specific construction that uniquely identifies any and all points on a line.

How can something have parts potentially?

First of all, this is again a falsehood. It doesn't 'have parts potentially'; you are, yet again, conflating the structure of the line to be divided with the structure of the operation of division. The line in question is a whole whose parts are potentially distinguishable by division. The only alternative to saying that this is true is to deny that it can be divided, which you cannot. Where you err is assuming that divisible (potential) is equivalent to divided (actual). This assumption is not made by the Aristotelian, (to get 'divided' you need an actual operation of division, so given that something is infinitely divisible, you can't assume that it is infinitely divided unless you can identify the particular operation of division that divided it). And thus everything you have said, which has assumed that what is divisible is divided, both begs the question against the Aristotelian and misses, as I said, the whole point of the Aristotelian position.

Treating 'divisible' as equivalent to 'divided' is modal collapse. 'Can be' is not normally equivalent to 'is', so there are only two possibilities on the table here:

(a) You are making a logical error. The fact that you keep getting the Aristotelian position even after repeatedly being told what it is, is a sign that this is the case.

(b) There is some principled reason why it has to be the case given the fundamentals of the field being talked about. You have given no such principled reason, because everything you have said assumes the modal collapse rather than explaining it.

19. A geometrical figure does not have parts. You may divide a geometrical figure in half or whatever, but a triangle or a tetrahedron in the geometric sense are shapes not actual things.

A ball of clay does not have two halves as parts, but may be divided to produce to halves of what together once made a ball.

Anyway, back to Zeno. If you walk at one meter per second, you will have been at 1/2 meters at 1/2 of a second, at 3/4 meters at 3/4 of a second and so on. No manner how far you go, you can always calculate the exact time you would have been at a certain location. It's never infinity.

20. None of you have refuted the paradox, which Bertrand Russell called very profound. First, you can make a line, which means that geometry applies to matter. Engineering works because of this. You can know that matter has infinite parts because i have shown there cannot be a smallest unit. We don't even have to seperate the matter in order to divide it. If it is divisible infinitely, and yet we don't seperate it, it then has infinite parts.

21. I will just add this question: how many parts are there of a ball? Think about that. Also I would follow, after you considered that, with the question "how many parts do those parts have". Matter is continuous, which means it goes on forever. Our minds say something must be either finite or infinite but matter alludes this characterization. Lastly, consider motion. In order to most from point A to point B you have to move half of it. If you moved half and you were at point A you didn't move half. For there to be distance there must be space, which means there are halves. But before you move that half, that half having parts/space, you have to move the second half, and so the cycle considers forever. Do we move? Yes. We don't know how though. I can do no more than this to show what Zeno's paradox has been all about this whole time

22. First, how is matter continuous? Are we ignoring the atomic theory?

Second, even supposing it was continuous, I don't see the problem here, in light of 19th-century rigorous mathematical analysis. Quite possibly I'm just missing something. But just as you insist (correctly) that there are infinitely many segments of which we must cover half, so there are infinitely many "befores" in which we can do it. So your paradox collapses. You say, e.g., "before you move that half, you have to move the second [?] half..." but this is all fine. We can subdivide time just as much as we can subdivide space, and so you will fail to reduce motion to a contradiction.

To be explicit, suppose we are moving one yard in one second. Well, before we can finish moving one yard, we have to finish moving half a yard. No problem. That happens at half a second. But before we can travel half a yard, we have to travel (i.e. finish traveling) a quarter yard. No problem. That happens at a quarter second. Etc. What am I missing?

23. Paradoxes aren't "refuted"; that's not how paradoxes work. One gives an account of them; that is all. You keep giving an account of it that commits what seems to be a logical error, as previously noted; you keep saying the Aristotelian account is wrong, but on grounds that are question-begging, as previously noted.

First, you can make a line, which means that geometry applies to matter. Engineering works because of this. You can know that matter has infinite parts because i have shown there cannot be a smallest unit.

You have not shown this; you merely keep asserting it while failing to show how you get the collapse of 'can be' to 'is' without committing the logical error this usually involves.

I will just add this question: how many parts are there of a ball?

It's a badly formed question; how many parts something has is relative to how, precisely, you are distinguishing them.

24. The motion would appear to be impossible both because the time would be infinite, as well as the space. Atomic theory was refuted when I showed there cant be a smallest unit of space. Modern theory deals in arithmetic solutions because doesn't help when you are dealing with infinite space that appears also to be finite. Finally, how many parts something has is not relative. The answer is that is goes on forever

25. Suppose GOD wanted to split a ball in half. Its done. Then He splits one of those into two. Then one of those pieces into two. He does this until it can't be done anymore. He places the largest piece on the left, then the next to the right, and the next to the right of that, and onward. Put your finger to the smallest part on the right side. Clearly there cannot be a smallest one because God would just divide it more and more. So it goes on forever. THEN God puts it back together. This shows that something spatially finite (meaning it has a beginning and end) can be at the same time spatially infinite. THAT is the paradox in a nutshell

26. Abstraction must still interface with one's explanatory terminus. The Past Eternal universe (...forever dividing...) gives the Non-Theist no traction here with respect to Actual, Potential, Time, and Change, as Abstraction soon leads one beyond Physics-Full-Stop (... http://disq.us/p/1o5v88h ...). As for Eternalism vs. Presentism, there too Abstraction soon forces one beyond Physics-Full-Stop (... http://disq.us/p/1pn13tf ...). One must move carefully here lest one (perhaps unintentionally) makes a move which claims the one must expunge in order to subsume (... http://disq.us/p/1or4mzh ...). The following two items allude to relevant layers here:

[1] http://disq.us/p/1pstind

~

27. I still don't see anything here, Anon, except equivocation over the definition of infinite. There's no "paradox" with time being infinitely divisible, any more than there is with the real line. Mathematicians have been very carefully analyzing these issues for over 150 years now, and objectors seem almost universally to just come in and say, "infinity, infinity, so paradox." Infinity is subtle and can't be treated so wrecklessly.

28. Things do not have potential parts. It has parts. Finite means there is a beginning and end. Infinity means there is no end. So the melding of these two opposites in time and matter is a paradox. A paradox can be solved and thus, yes, its difficulty "refuted". Zeno's paradox has never been solved and I don't think it ever will. It is too straight forward. I think it is a beautiful paradox and shows how limited our vision of truth can be. In calculus something finite can also be infinite. That is not a problem if it is not talking about infinity of space. Another paradox is the circle. Take a line. It has a definite length. Then make it into a circle. Suddenly its length is factored by pi, so the length is no longer definite. That is a huge paradox

29. You can have a line with a finite length, and it will still have an infinite number of points. The line has ends, but the enumeration of its points does not. You are trading like mad on this equivocation.

I have no idea what you mean about the circle. A circle has definite circumference.

30. Its is not an equivocation. The same segment is infinite and finite at the same time. Also, a circles circumference is pi-r-squared, which is an infinite size

31. OK. We'll leave it there I think. Thanks.

32. "Then make it into a circle. Suddenly its length is factored by pi, so the length is no longer definite. That is a huge paradox"

The length of a circumference is as definite and well-defined as the length of a line.

"Also, a circles circumference is pi-r-squared, which is an infinite size"

"pi-r-squared" is not an "infinite size". not here, not in China, not in Mars or anywhere in the known universe or outside of it, with the possible exception of the insides of very confused minds.

33. The number pi goes on forever. Even though there are ever increasingly small segments, taken as a whole the length is infinite because the number never ends.

34. @Anonymous:

"The number pi goes on forever. Even though there are ever increasingly small segments, taken as a whole the length is infinite because the number never ends."

This is complete rubbish, please do not clutter the combox with it. The number pi is a real number, and therefore finite, and it is not even difficult to find majorants and minorants (circumscribe and inscribe the circumference in squares; do the math).

Possibly what you want to say (but here I am guessing) is that for the number pi (as for any irrational number) its decimal expansion is an infinite, non-periodic sequence, but there is nothing mysterious about this. Since by definition, a decimal expansion is a specific infinite sequence of approximations to the number built out of rational numbers with powers of 10 as denominators.

You are very confused; you simply do not know what you are talking about.

35. Nothing you wrote contradicts what I said. The infinite sequence means an infinite length

36. The infinite sequence you speak of is spatial in a circle

37. @Anonymous:

"Nothing you wrote contradicts what I said. The infinite sequence means an infinite length"

I have walked down this road often enough to smell an ignorant crank miles away and know the utter futility of reasoning. I suggest you take a class in elementary analysis and then come back. Peace. Exeunt.

38. There is a "limit" in a circle but the termless sequence goes on forever, so how can there be a limit? That is the paradox. If you figured this out you would deserve a math award, but alas you can't

11. There can be five apples in mind-independent reality, or five dogs, or five donuts, but not five itself, considered as a kind of freestanding entity.

I have always wondered about this. There must exist a common physical property in the five apples, and in the five dogs, and in the five donuts. For it there weren’t that common physical property then it wouldn’t be possible that that we (or a computer for that matter) count up to five and stop. But if there is such a physically instantiated property of “fiveness” why believe that five is a mind-dependent freestanding entity?

If there is no reason then what exactly is the difference between concrete and abstract? Why is the triangle I draw on a piece of paper concrete, but its physical property of “triangularity” abstract? I understand it is a more general property belonging to many things I might draw on a piece of paper or mold with a piece of clay or detect is some crystal formation, but why call triangularity an *abstract* property? And not simply a physical property belonging to many physical objects, such as, say, electrical charge?

To put it differently. Suppose I hold that geometrical objects are the common physical properties of what I draw on a piece of paper using pencil, rule, and compass. And I hold that geometry is the science about such physical properties. What would I be missing?

Mathematics… is an enterprise consecrated to the ideal of precision.

Mathematics works with rules, and rules are by definition precise. There is no mystery here.

Modern physics is very difficult indeed; unlike mathematics, it is concerned with the physical world.

No, modern physics is concerned with physical phenomena. If the physical world which produces the physical phenomena we experience turned out to be very different than how we imagine it (such as, say, that we all live in a computer simulation), not one iota would have to changed in all our books of physics, chemistry, biology, neuroscience, etc. By discovering the order within physical phenomena modern physics cannot tell us how reality is but only how reality surely is not. For reality cannot be such as to *not* produce that order.

1. Abstraction, Reference Frame, and Perception:

You make good points all around. As Dennis Bonnette points out, “…....perception-dependent judgments and calculations cannot possibly be ontological in nature….” The "interface" as it were between [A] the fundamental and self-explanatory nature of reality and [B] the contingent Reference-Frame of the Contingent Being must coherently traverse the affairs of Solipsism and Idealism – and leave them behind – before arriving at any justifiable *Ontic* claim. Very briefly: http://disq.us/p/1o5v88h

Also, from another direction, as another commented elsewhere:

“…a solipsistic universe would be indistinguishable from a theistic universe…”

It is not so much “Triangularity Is Abstract But Tree Is Concrete” but, rather, it is the whole show from A to Z which is in play with respect to *ontic* claims (….for examples of just who real the problem is one only needs to survey the discussions amid Eternalism and Presentism.…).

2. Here’s the first of those two parts described in the previous comment:

I didn’t mean that you claimed Kreeft makes Time his A – Z and I should have been more clear on that. Rather, I mean that your body of premises aimed to argue against the proverbial A.T. Meta Proofs treat Time that way. As in:

The reason that Time is, so far at least, your A and your Z (so far) is that your entire analysis has discussed change from the perspective of contingent frames of reference. Which is to say you’ve not even discussed whether or not temporal becoming is real or illusory. That is to say you’ve not gone far enough upstream or downstream with whatever your own paradigm’s explanatory termini happen to be to discover whether or not the A.T Meta’s explanatory termini are in fact coherent or not.

Part of the problem which “Change In The Universe” faces is [A] it’s irrelevant to the proverbial Proofs vis-Ã -vis change which is why Feser and others are happy to grant the past eternal universe and [B] whether or not it (temporal becoming) is even *real* or *actual* in the first place. If it is real (actual), well then we are back to [A]. If is not real (if it is non-actual), well then we are back at [B] and Eternalism / Presentism arrive on scene. Therefore, Time is left just dangling in midair, as if it were the A – Z of one’s entire T.O.E.

To address Change without addressing Temporal Becoming is to address Change without addressing Time – and to stop “there” is to fail to address God, or, rather, the Divine Mind as per the Christian metaphysic. Not only that, but it also commits you to Time as your terminus of explanation on all points in this entire discussion of Proofs – which is – in the end – to define one’s T.O.E. by this or that contingent Reference Frame – which as pointed out does not go far enough and which is – in the end – irrational.

The Self-Explanatory comes in and by the Absolute's Frame of Reference, which cannot be less than Self-Reference. Where and how the Trinitarian Life subsumes all points there is another discussion but it is worth pointing out that the curious affair of short-circuiting reason's demands for lucidity within contingent reference frames just won't do.

3. Here’s the second of those two parts:

**Clarification:

I stated “The Self-Explanatory comes in and by the Absolute's Frame of Reference, which cannot be less than Self-Reference. Where and how the Trinitarian Life subsumes all points there is another discussion but it is worth pointing out that the curious affair of short-circuiting reason's demands for lucidity within contingent reference frames just won't do.”

However, by “short-circuiting reason's demands for lucidity within contingent reference frames just won't do” I am not referring to reason’s demands for lucidity within contingent reference frames, for, obviously, Reason in her proper role as truth-finder demands lucidity “there too” so to speak. Rather, what the comment references is Reason’s demands for lucidity not “just within this box” (so to speak) but in fact she reaches outward into all possible reference frames – to Totality. But of course Time is neither Absolute nor is Time The Absolute’s Own Reference Frame. Therein Physics as such leads the rational mind to that which is beyond physics – just as – in the Contingent Being – the contingent affairs of reason itself leads the rational mind beyond itself and into Reason Itself.

As Dennis Bonnette points out, “…....perception-dependent judgments and calculations cannot possibly be ontological in nature….” Any contingent frame of reference necessarily fails to be self-justifying. As in self-explanatory. Any such Reference Frame leads one beyond itself, and into the Absolute's Reference Frame which cannot be other than Unconditional Self Reference amid the Infinite Knower, the Infinitely Known, and all Communique therein, as the triune topography of Infinite Consciousness – of the Divine Mind – presses in ( … https://www.metachristianity.com/thoroughly-trinitarian-metaphysic/ ..). With respect to Physics, Time, and our own reason vis-Ã -vis our own contingent reference frame:

[1] http://disq.us/p/1o5v88h
[2] http://disq.us/p/1pn13tf
[3] http://disq.us/p/1or4mzh

The reality of temporal becoming is in fact actual or else it is in fact illusory. If the later: well then there is that discussion and the Non-Theist must deny change all together (...the discussions amid Eternalism, Presentism, and so on...). If the former: well then we arrive at Feser's (and others) seemingly odd willingness to just GRANT the Non-Theist his past-eternal universe for, as that odd willingness tells us, it grants *no* relief to the troubles which Non-Theism faces amid temporal becoming and the causal ecosystems therein.

12. A segue of sorts: http://disq.us/p/1ptuwhq

13. Did you ever have the chance to meet Wallace when he was at Pomona, Dr. Feser? His work was still talked about when I was at Claremont

14. My previous comment linked to a segue of sorts. The comment was one in a series which challenged the author's arguments against some standard proofs of God. In it I linked to this OP by Feser as per the following comment which is not visible there now:

It is claimed that radioactive decay presents a problem. That is fine as far as it goes. However, to defend that one will need to address the three sources of Feser which refute that claim, namely the two blog posts (...linked to earlier...) and, then, if one has a digital copy of Five Proofs one can search for the word "radio" or "radioactive" and, the several hits which one finds will also have to be addressed. As for what is called word-salad, that is a discussion on the wider arena surrounding causation and change which have to be -- at some point -- "pulled in" as it were and shown to "fit" or else "not fit" into the argument. That is true for any argument, not just Non-Theistic or Theistic. Sure, expanding the lens of this discussion to point out that "this box" cannot be coherent if it fails to hold up "further out" (so to speak) may seem like "world salad" to some, but that is a mark of one's unawareness of why and how any particular "narrow slice" of observational reality is but a part of much larger, wider, discussions. And those discussions are not mine, but are in fact held by both Non-Theistic and Theistic thinkers alike in what is a rather interesting array of descriptions of reality's fundamental, or irreducible, nature. The fundamental nature of Time, Change, and Becoming are in fact inter-related (...perhaps intra-related is better...?) and are in fact relevant to this discussion and, therefore, all the affairs of "Reference Frame" necessarily weigh in.

A segue of sorts comes through observations of Abstractions and the Empirical: http://edwardfeser.blogspot.com/2018/01/david-foster-wallace-on-abstraction.html

The Wider, Thicker "metaphysic" surrounding radioactive decay cannot be defined as "word salad" if one wants to address the *actual* definitions (...premises...) put forth by the Christian metaphysic. To discount such definitions and premises in that manner leads one to misread the actual claims and to then embark on presenting arguments against this or that straw-man. Hence the problem of radioactive decay, causal ecosystems, temporal becoming, reference frame, and so on.

15. "Much more on these subjects forthcoming in the philosophy of nature/science book I am currently working on."

Awesome. Do you have an ETA? Also, will it be accessible for non philosophy majors?

16. I’m a philosophy student - any recommendations on seminal works that either defend or critique Platonic Forms? As a Catholic I’m interested in theological dimension of the debate as well.

17. "Abstraction by its very nature leaves out much that is in concrete reality, and the more abstract the model arrived at ..., the more that is left out."

This sums up my thoughts on A-T abstractions like final cause, act/potency, per se/per accidens cause and effect, etc.... -- They're abstractions that by their very nature leave out much of reality which nevertheless are then manipulated to draw absurd metaphysical conclusions about ultimate reality.

1. >This sums up my thoughts on A-T abstractions like final cause, act/potency, per se/per accidens cause and effect, etc.... -- They're abstractions that by their very nature leave out much of reality which nevertheless are then manipulated to draw absurd metaphysical conclusions about ultimate reality.

He is back! Djindra has abandoned reason and has returned to his roots as an intellectually inferior Gnu troll!

Here is rational response to his nonsense AT abstractions by their very nature leaves out much that is in non-metaphysical & general philosophical descriptions of reality.

Of course one would be a fool to claim "Philosophicalism" would be any better then Scientism.

Naturally nobody here is claiming A-T Philosophy is exhaustive in explaining reality unlike the Scientism advocating boobs Djindra & his Gnu buddies idolize.

18. I am not keen on the anti-abstract object/anti-Platonist virtue signalling either. Theists are as committed to transcendent universals (in their case as Divine intentional objects) as Platonic naturalists. If it comes to a battle it’ll be between those two postione with old basic Aristotle left in the dust.

Of course the point of such rhetoric is to stress that abstract particles which we only know from their causa roles are actual entries with irreducible dispositions (they have little else!) not merely structural features in a mathematical modal (theory).

19. This kind of abstraction explains things like the Numberphile video saying that 1+2+3+4...= -1/12
Mathologer had a great response video explaining that this is a misunderstanding of analytic continuation, a way of extending a function beyond its domain. It's like taking statistical data collected over a period of time and extrapolating it into the future. And it can lead to statements like "given this rate of demographic change, by 2100 we expect that 130% of the U.S. population will be hispanic."

Obviously, the -1/12 answer is the result of purely analytic reasoning applied to an abstract concept. It has nothing to do with reality.

20. Note everyone agrees with Aristotle's counterarguments. See Cantor's arguments vis-a-vis transfinite sets.

1. Cantor's entire case is dependent on naive set theory, which was shown to be inconsistent. The axiomatic set theories which replace it are based on question-begging axioms.

21. Hi Ed,

News has it that Germain Grisez has passed. I know you sometimes write about philosophers who have died in honor of their memory.

22. 1 + 2 + 3 + 4 + ... is said by some to equal -1/12. This may be true if we stretch the meaning of equal and we do our calculations in a complex plane.

Supposedly this "equality" is used by superstring theorists to justify the 11-dimensional spaces within which their strings wriggle about.

Is this a case of the abstractions of mathematics hinting at abstractions of physics that tell us something about how the world works?

Or is this the case of an abstraction in math just happening to coincidentally line up with an abstraction in physics?

1. The problem is not with equality, but with the assumption that the Riemann Zeta function describes what happens when you add sequences of numbers together. This is precisely what you abandon when you use analytic continuation.

2. I'm less concerned about the equality than I am about how the equation "works" in both the abstract realms of both mathematics and physics.

The use of analytic functions in working with physical quantities makes sense to me in that analytic continuation guarantees very smooth (i.e. potentially realizable in the real world) functions.

However, if the Riemann Zeta function has no analogue in the "real world" then it is just happenstance that the "equality" works for superstring theory.

Also, it seems as if this "equality" is used for normalization, i.e. eliminating infinities from physics equations.

It all seems quite sketchy to me.

23. Part 1

….this line is finite in that it has ends, but it is also infinite because one can keep slicing its parts into still more parts, ad infinitum, such that in fact there is no end to its parts and the line is, thereby, infinite…

There are several problems with this. First, as a Christian, it is not a problem to infer that the proverbial Beneath & Above or the proverbial Always & Already is not only “infinite” (…perhaps Absolute is metaphysically more precise…) but in fact interfaces with the Created Order such that – at some “ontic-seam” somewhere we run into that “phase change” or “transition” so to speak amidst the Finite and the Infinite. It is not “that” which is the “problem” here. Rather, it is along the following lines where “Zeno’s Dichotomy Paradox” runs into trouble:

Time, Space, & God – a brief look from http://edwardfeser.blogspot.com/2018/02/time-space-and-god.html

"So, there just is no sense to be made of the idea that there is something distinct from God that he cannot not create. If he cannot not create it then that is only because it cannot not exist, in which case it is purely actual and subsistent being itself and thus really identical with God. If it is really distinct from God, then it is not purely actual or subsistent being itself, and thus it can fail to exist and God can refrain from creating it. The supposed middle ground position between pantheism on the one hand, and affirming the contingency of time and space on the other, is an illusion."

We start a bit downstream and work our way upstream, towards that “metaphysical wellspring of all ontic possibility” as we are ever so careful with respect to the trajectory of that the proverbial Arrow of our semantic intent:

Potential Parts, Semantic Intent, & Reference Frame:

First, a brief quote from http://disq.us/p/1oejdyo which alludes to several transcendentals, the traversal of which all Non-Theistic termini fall short of:

Actually, there isn't any ontic-difference because the Change-In-Perception between the Now Self and the Future Self and the Past Self does not happen. You are actually a "Part" of the Block, which has many "Parts", even as you are, simultaneously, sort of hovering outside of it and looking down / over / across it and seeing different slices through different frames of reference. As the Conscious Observer your perception changes even as your perception never changes because there is no change nor anything to Q or Cause said change in your Perception as the Conscious Observer. "You" as the Conscious Observer are "simultaneously" both a Static / Motionless "Part" of the Block and an Observer hovering outside of the Block looking down / over / across it and seeing different slices through different frames of reference. Which carries us full Circle as we repeat the cycle given that as the Conscious Observer you are, now, even still, a static / motionless "Part" of the Block "such that" your perception never changes, ,and, also, you are hovering / moving "such that" your perception changes as you look across / over different slices of the Block, because there is no change nor anything to Q or Cause said change in your Perception as the Conscious Observer. Your perception as the Conscious Observer never changes even as your perception as the Conscious Observer changes as you observe things from different frames of reference. Got it? Don't ask questions. Just believe.

End quote.

Continued....

24. Part 2

Akin to the debate amid Eternalism vs. Presentism, there is a concrete sense in which (…contra physics-full-stop…) it is the case that *both* are actual in the Christian metaphysic. That is found only in and by the affairs of Timelessness, Pure Actuality, and ALL reference frames whether Possible or Actual vis-Ã -vis the Divine Mind, Possible Worlds, and so on. That is why we find that Physics, when followed to all possible termini, leads us out and beyond herself, beyond Physics-Full-Stop, rather than in the illusory shadows of non-being forced by all Non-Theistic termini. Eternalism and Presentism both subsist but, of course, when the Christian makes such a statement his semantic intent is quite different than that of the Non-Theist.

Then, just the same, that is why we find that reason itself in and by all Non-Theistic termini lands not in the convertibility of the necessary transcendentals with respect to *being* but, rather, in the illusory shadows of non-being. Is that Presentism? Is it Eternalism? In fact it is again both (…by the Christian’s semantic intent and not by the Non-Theist’s…) for with respect to our own “contingent reason itself” so to speak it is the case that, like Physics, so too reason itself when rationally followed leads one beyond one's own unavoidably contingent reason and into the Necessary & Irreducible vis-Ã -vis Reason Itself. The Divine Mind presses in (... http://disq.us/p/1o5v88h ...).

The Self-Explanatory:

The Self-Explanatory comes in and by the Absolute's Frame of Reference, which cannot be “something less” than the Totality of Unconditional Self-Reference. Where and how the Trinitarian Life subsumes all points there is another discussion but it is worth pointing out that the curious affairs of all possible Non-Theistic explanatory termini are forever short-circuiting reason's demands for lucidity within contingent reference frames, and that just won’t do. Now, obviously Reason in her proper role as truth-finder demands lucidity “there too” so to speak. Rather, the semantic intent there aims at the fact that Reason’s demands for lucidity cannot be rationally held if doing so forces one to abort everything but various contingent contours “just within this box” (so to speak) but, rather, it is the case that Reason reaches outward into all possible reference frames – to Totality.

Space-Time, Reference Frame, & Little Boxes:

But of course Time is neither Absolute nor is Time The Absolute’s Own Reference Frame. Therein Physics as such leads the rational mind to that which is beyond physics – just as – in the Contingent Being – the contingent affairs of reason itself leads the rational mind beyond itself and into Reason Itself. Again the Divine Mind presses in. As Dennis Bonnette points out: “…....perception-dependent judgments and calculations cannot possibly be ontological in nature….

Any contingent frame of reference necessarily fails to be self-justifying. As in self-explanatory. Any such Reference Frame leads one beyond itself, and into the Absolute's Reference Frame which cannot be other than Unconditional Self Reference amid the Infinite Knower, the Infinitely Known, and all Processions / Communique therein as the triune topography of Infinite Consciousness – of the Divine Mind – presses in ( https://www.metachristianity.com/thoroughly-trinitarian-metaphysic/ ).

Semantic Dependence, Abstraction, Mind, Reference, & Zeno’s Dichotomy Paradox:

In the com-box from http://edwardfeser.blogspot.com/2018/01/david-foster-wallace-on-abstraction.html we find various hints at something akin to this:

….this line is finite in that it has ends, but it is also infinite because one can keep slicing its parts into still more parts, ad infinitum, such that in fact there is no end to its parts…

In that discussion...

Continued...

25. Part 3

In that discussion some of our Non-Theist friends are claiming that the shuffling of parts in, say, this or that universe presents a paradox – that of “Potential Parts”. Obviously if one refers to the Milky Way as A-Thing, one's syntax is referencing X. If one then refers to the Milky Way as "All-These-Things" then again one's syntax is referencing, not X, but Y.

In E. Feser’s “From Aristotle to John Searle and Back Again: Formal Causes, Teleology, and Computation in Nature” we find several segues here:

"The technical sense in question is essentially the one associated with mathematician Claude Shannon’s celebrated theory of information. Shannon was concerned with information in a *syntactic rather than *semantic sense. Consider the bit, the basic unit of information, which has one of two possible values, usually represented as either 0 or 1. To consider a bit or string of bits (e.g., “11010001”) in terms of some interpretation or meaning we have attributed to it would be to consider it semantically. Semantic information is the sort of thing we have in mind when we speak of “information” in the ordinary sense. To consider the properties a bit or string of bits has merely as an uninterpreted symbol or string of symbols is to consider it syntactically. This is “information” in the technical sense. When instantiated physically, a bit corresponds to one of two physical states, such as either of two positions of a switch, two distinct voltage levels, or what have you..."

We run into this same array of affairs vis-Ã -vis semantic intent when we speak of the triune topography of Divine Simplicity amid Trinity, God, the Divine Mind, and so on. Another slice of this pie which we have to keep in mind is the fact that, as Ravi Zacharias discusses, men knew that three fish were not the same as one fish in the first century and yet they converged within the syntax of 1 / 3, and so on. The distinction of using a term in a different sense with respect to [A] vs [B] is something we can miss or perhaps downplay. Yet Zacharias goes on to point out that language employed in reference to such metaphysical topography arrives on scene:

[1] univocally and
[2] equivocally and
[3] analogically.

In several approaches we can add two more:

1. the fact that distinction is not to be equated to division and
2. the fact that "distinction-void-of-division" is not incoherent.

Care must be taken to note in which sense our various semantic intents are constructed as we find, yet again, that permitting frail and contingent reference frames to define one's T.O.E. is inexplicable. And ultimately irrational. It sometimes seems that we really do not comprehend the sheer Totality of our dependence upon the unavoidable Ontic Arrow of the Absolute's Own Reference Frame whether we are traveling Downstream or Upstream.

Wittgenstein–esc indeed, only, the Christian refuses to settle for a half-narrative. As per [1] http://disq.us/p/1owou01 and [2] http://disq.us/p/1owovkf Gilson asks, “When God created space, where did He put it?” The very syntax of “space” and all contours of that very concept are themselves but frail and contingent reference frames.

Once again we find that the Absolute’s Own Reference Frame necessarily sums to Unconditional Self-Reference and, also, we find that the Non-Theist misses the unavoidable fact that the Absolute houses a fundamental relationship with — not SOME — but ALL frames of reference whether we reference Possible Reference Frames or Actual Reference Frames.

We'll leave our Non-Theist friends to unpack within the frail and contingent mind of the contingent conscious observer what they suppose [A] Absolute Self-Reference and [B] Infinity and [C] the Trinitarian Life all share in common vis-Ã -vis the *Christian's* metaphysic.

scbrown(lhrm)
~