Long ago I explained why All Godless Universes Are Mathematical and how physicalism already explains The Ontology of Logic and how numbers are not the same thing as quantities but only codewords that refer to them in How Can Morals Be Both Invented and True? And I’ve already explained why physicalism really means that every possible thing is reducible to some physical (as in, nonmental) thing in Defining Naturalism: The Definitive Account. But in my recent discussion of all this with Alex Malpass one thing we skipped for time was how actual infinities already exist everywhere. Which also raises other questions about how I think other weird mathematical objects exist. So I’m filling that in here.
The Background
I ascribe to a version of nominalism, the idea that all mathematical words and symbols and equations are just names for patterns of things out in the real world. And I am more particularly a modal structuralist, whereby all mathematical language describes actual or possible structures, whether causal structures or geometric structures or objects with structure and so on. All math is therefore just hypothetical discourse about possible realities. I’m oversimplifying, but getting more nuanced isn’t necessary here, unless you want to ask in comments. Generally, though, anytime you want to know my position on common philosophical disputes you can see How I’d Answer the PhilPapers Survey (and, for the questions they later added, How I’d Answer the 2020 PhilPapers Survey; though the first has my take on nominalism), and you can ask questions in comments there for anything not covered, or for clarification on anything that is covered. But here, let’s talk some Ontology of Mathematics (a subset of the Ontology of Logic).
Malpass took the position that nominalists require the physical (actual) world to have infinite resources. In particular he gave the example that unless the universe is infinitely large, we cannot explain actual infinities or how they work or how we can study them, because there can never be an actual infinity of anything. But he overlooked two things.
- First: as I’ve often explained, not everything has to actually exist in nominalism. It only has to potentially exist. Thus we can make sense of an infinitely large world and derive conclusions about it because we can easily describe how it is a potential outcome of any actual world. Any space can be expanded endlessly, producing an endless amount of it. If you have one of a thing, you can have two. And if you can have two, three. And so on to every infinity. This is basically literally what Cantor did when he worked out how different cardinalities of infinity exist—because they “exist” as the potential rearrangements of any actual things. They did not have to exist as actual things for him to work that out, or to work out what the differences across infinities thus were. Hence we don’t have to actually make a thing to work out what would be the case if we did make it, just as we don’t have to actually grow a unicorn in a lab to know it will have trouble getting through our kitchen’s doggy door. Nominalism is always about hypothetical structures—and thus what those would look like, what properties they have, if realized. We don’t have to realize them to do that. But they do have to be realizable (in principle—as in, we can describe the resulting realization). Aristotle already made this point thousands of years ago when he invented nominalism. So we mustn’t overlook it as a fundamental feature of any effective nominalism.
- Second: we don’t need a universe to be infinitely large for it to contain actual infinities anyway.
The latter point should not distract you form the former point, however.
Just because it happens to be the case that any finite collection already has infinite resources doesn’t mean we need that to be the case. Even were it not the case, potential infinities still exist in every required sense, and so all transfinite mathematics requires no further resource than hypothesized infinities made out of finite resources. Like infinitely duplicating a rock in your back yard. We don’t have to actually do that to work out what would be the case if we did: because that is inherent in the hypothesized state, and thus requires nothing more than that. This differs from straight fictionalism (another kind of nominalism) in that fictionalism imagines this has nothing to do with reality at all, whereas modal structuralism imagines that it is always anchored to physical reality in a crucial sense: every mathematical structure has to be physically realizable, using things that already do or could exist. It is thus an exploration of possible realities and constrained thereby. Thus every mathematical statement must be realizable in some physical structure to “mean” anything and thus be analyzable into anything we would call “knowledge.” Math is just hypothetical physics. So any world with any physics already contains all mathematical truths. Even a state of nothing does.
And that’s why physicalism fully explains why maths and logics can be developed that work and thus happen to so successfully and reliably describe reality. Any real thing is reducible to a mathematical description of its structure. Therefore every real thing in every possible universe is inalienably mathematical. There is no other way to have it exist and not be mathematically describable down to every necessary detail. Which means the converse is then true: any fictional mathematical structure will correspond to a world that can potentially be realized in the same way. There will be, in other words, some corresponding physical structure to every mathematical statement, such that that statement can never be realized without realizing the physical structure it describes. But we can still think about it even when never realized. Because that’s what we built math for. And the equivalency here is total, meaning, that is the sum total of the meaning of all mathematical statements. And therefore that is all math is doing: walking through all the potential rearrangements of the furniture of the world. Which requires nothing but any real world and the potential for it being physically rearranged. Platonism has nothing to do here. It becomes just angels pushing the planets—after Newton already proved the existence of gravity. We simply will never have any need of that hypothesis. While physicalism does all the work required.
Hence the title of this article is how all math is real—as in, in what way it is real. It is not that all possible mathematical objects actually exist somewhere; but rather that all possible mathematical objects potentially exist anywhere. Every mathematical object describes some physical object—and only carries any meaning at all insofar as it does. This does not require that the physical object described actually exist (yet or ever). It only requires that it could, as in, the object described, if fully realized, would be solely and entirely physical (geometry, quantities, ratios, causalities and all). This does suggest God may be logically impossible. But we have not yet proved physicalism true to a logical certainty, so we cannot be certain God is logically impossible. But we do have grounds to suspect it (see The Argument from Specified Complexity against Supernaturalism and The God Impossible).
Some Examples
You might say, well, “I have never encountered 326,519,438.004 objects” (although you just did: in your entirely physical, reducibly nonmental brain) “nor is there anything physical about it” (yet there is: the word refers to a potential physical fact: a specific ratio of things, which can be realized by simply adding more things to any smaller set of things, including some fractions of a thing as well, until you have the quantity that that number both refers to and can operationally substitute for in any language). Being a fraction (with that “.004” tacked on), it refers to fractional quantities, and fractional quantity is a fundamentally physical property: space, time, matter and energy all by definition possess it. It can refer to some actual fact (odds are, such a ratio exists between the respective lengths of some two objects in this universe—because there are so many objects in this universe to stand in ratio to each other) or a hypothetical fact (such a ratio can exist between two reducibly nonmental objects, without requiring anything irreducibly mental, e.g. I could cut two wires right now that have that ratio between them, just as I could collect 326,519,438 meter sticks and then add one 250th of a meter stick). And that is simply what that number means: the actual or potential physical ratio or fractional count of quantities it signifies (and thus can stand in for).
By analogy, there are no unicorns, either, yet the word “unicorn” still refers to a hypothetical physical fact: unicorns really exist whenever the physical (and thus reducibly nonmental) entity described by that word exists. And if they don’t “really” exist, they exist “fictionally,” since the thought of them exists in the brain that thinks it. And as long as the brain doing that is reducibly nonmental, then by the law of commutation, so is “the idea” of a unicorn. And even when no one has ever thought of it yet, the possibility of a physical arrangement of a physical brain corresponding to thinking it physically exists everywhere there is. And nothing more is needed than that. Just places to be, and what you could build there (see The Argument from Non-Locality). It’s not like if we flipped a switch on a strange machine that “turned off” or “erased” the Platonic realm we suddenly couldn’t think of a triangle or make one, as if some invisible force stopped us. No. Just having a brain that can compute things and a space to build them in is enough to have everything there is about triangles be true. Flipping that switch would do nothing. And so no Platonic realm is needed to explain anything at all about math or logic, or even ideas.
Thus if I have a gold ring, a gold cube potentially exists. Because I can mash it into a cube. But I require no supernatural power to do that. Nothing irreducibly mental need exist for a ring of gold to be potentially a cube of gold. Thus, potentially existing things are not irreducibly mental and thus not supernatural. Hence the same follows for a ratio like 326,519,438.004: if not referring to an actual ratio (like two actual wires, or the ratio of distances between two stars as measured from a third), then it refers to a potential ratio. And potential things neither are irreducibly mental nor require the irreducibly mental (as I discussed decades ago in Sense and Goodness without God, pp. 125-26).
This is why anyone who says anything like, “There is nothing physical about the fact that 4 is more than 3” is simply wrong. The fact that four sticks physically includes three sticks and another stick is a physical fact. And likewise any other instantiation of this pattern of physical quantity (of “four” things). The fact that this property can be shared by many things (indeed infinitely many: counts of apples, diameters of stars, areas of deep space, weights of rocks, eyes on a planet-eating space amoeba, etc.) does not require anything to exist but physical objects. In other words, there is no immaterial “thing” that you can “remove” from the universe and somehow make it impossible for “four sticks physically includes three sticks and another stick” and “four stars physically includes three stars and another star” from both still being true (and so on for every other possible expression of the same or any quantity).
Hence we invented words to refer to that repeating pattern (which is a physical pattern, always and ever, even when only about hypothesized physical systems and not yet actual ones), which is, in this case, simply that “4 is more than 3.” That sentence did not exist a million years ago (humans invented it). Only what it refers to existed a million years ago. And what it refers to is the physical fact that any four things includes three things and another thing. Hence it refers to nothing else but the repeating physical pattern of “four physical things includes three physical things and another physical thing” (regardless of what kind of “thing” these are).
And Now for Infinities
One of the odd things we’ve discovered is that there are infinitely many infinities, each larger than the next. These are called “cardinalities” of infinity and are named aleph with a number, starting with zero (or “null”) for the smallest possible infinity, which is just the set of all countable quantities. But how can even that infinity “actually exist” (not even just potentially, but actually) in a finite world? Much less even bigger infinities? Well, prepare your mind.
Aleph-null = There are, right now, infinitely many geometric points on your fingernail. You don’t have to count them for there to already physically be an infinite number of them, right now. There are therefore infinite physical resources on your fingernail. Quantum mechanics makes no difference to this fact, as the functional fuzziness of spacetime at quantum scales makes no difference to the geometric properties of even those fuzzy spaces. There remains an actual infinity of geometric points on your fingernail. Physically. Actually. Right now.- Aleph-one = Then we add irrational (uncountable) quantities. Are there infinitely many irrational quantities on your fingernail right now? Yes. We could carve up your fingernail into all possible triangular areas, which thus include triangles with a ratio of sides equal to pi and every other irrational division of space. We do not have to actually draw these triangles: they are already there. Just as we do not have to count the geometric points to know there are, right now, infinitely many of them, so we do not have to draw every possible triangle on your thumb to know there are uncountably infinitely many of them. Those spaces exist. And there are uncountably infinitely many of them. Physically. Actually. Right now.
- Aleph-two = Then we add permutations. So, if we draw an arrow from each of those triangles to some other triangle, there are infinitely many ways to do that—i.e. there are infinitely many ways to arrange all possible triangles on your fingernail into a sequence. Again, we do not have to draw these arrows, just as we don’t have to draw the triangles or count the geometric points. The sequences are already there, right now, on your fingernail.
- Aleph-three etc. = Then it’s permutations all the way down. Just rinse and repeat: draw arrows from one complete sequence to another complete sequence and there are infinitely many ways to arrange those sequences. The quantity of all possible sequences of all those possible sequences of triangles on your fingernail is aleph-three. All possible sequences of all possible sequences of all possible sequences of those triangles is aleph-four; and so on.
- Aleph-aleph = There are infinitely many permutations of permutations of sequences of triangles on your fingernail. Therefore your fingernail right now physically contains all possible infinities. And that’s even before we include potential structures. Even the finite space of your mere fingernail contains infinite things, and thus infinite physical resources to ground all mathematical truths about infinities. Physically. Actually. Right now.
You can say all the “actual” arranging here—the actual counting, the actual drawing of arrows—only potentially exists (no one has, or right now even could, do those things). But the counting and drawing is incidental. If there are three melons, there are three melons, regardless of whether you have counted them or drawn every possible sequence of arrows between them. So the count of melons really exists, and physically exists, as a quantity, even if no one ever counts them. Likewise their surface areas, and thus the sum of their surface areas, exists whether anyone measures them—–fully, physically, actually exists. Likewise there are six different ways to arrange those three melons (six different ways to draw arrows between them), even when no one has yet drawn any of those arrows. Every sequence of those melons exists, right there, all the time. Their very existence comes with the existence of every possible sequence of them. The actual arrows might not be drawn in the sand beside them, but you can stare at those spaces and realize every possible arrow is there. Drawing the arrows does not make something physically exist that wasn’t already there. You are not a sorcerer. When you draw different sets of arrows between those three melons you are simply acknowledging the physical, actual existence of those directions already being there, and you are just labeling them. So those six arrangements of those three melons actually exists—and could not but exist, the moment the melons do. And we need nothing more than the physical existence of hose melons for all that to be true.
Therefore, as for any three melons, so for infinitely many triangles on your fingernail: all actual infinities exist there, all the time—and could not not exist there. There is no way to have a fingernail and not have infinitely many spaces on it or there not be infinitely many ways to connect them. So all actual infinities—the whole infinite quantity if infinities—exist on your fingernail, indeed exist in every finite space. All worlds therefore have infinite resources to realize all possible mathematical structures. So this is not a problem for nominalism—at all, least of all modal structuralism. And likewise every other mathematical object. Some perhaps are not physically realized (like, say, an actual hundred-dimensional sphere) but all such are potentially realizable. All a hundred dimensional sphere is is a sphere with more dimensions added to the ones we already have, which we can physically describe, and which is potentially physically realizable—indeed, all “a hundred dimensional sphere” means is a physical object, a sphere, spanning a hundred dimensions of space, whether that space or that sphere ever actually exists or not.
And likewise every mathematical object.
- Imaginary Numbers are physical descriptions of rotation in space (which we observe actually physically realized in electromagnetic systems).
- Fermat’s Last Theorem describes any physical objects with sides in the ratio of an + bn = cn and the reason no such object with any whole measure above 1 can ever exist with n > 2 is that physical geometry makes all other such shapes impossible. For example, 23 + 43 = 8 + 64 = 72, and the cube root of 72 is ~ 4.16, which is not a whole quantity but fractional. And there just is no way to twist up the geometry of any physical space to get it to be a whole quantity—you always need “an impossible kind of elliptic curve,” a physical shape that can never be realized.
- The Monster Group is just a description of a particular physical arrangement of a certain kind of system.
- Number Theory is just a study of the possible physical arrangements and relationships of discrete quantities.
- And so on. Markov Chains. Probability Theory. Octonions. Hilbert Space. Prime Spirals. Euler’s Formula and Identity. Always some physical system is being described (whether causal or geometric or anything else)—often one by now actually realized, but when not, always potential outcomes of physical procedures that no person or system has yet undertaken or undergone.
And likewise for every fiction, every possible world—like a world populated by unicorns, or solely filled with shrimp, or containing only one shoe, or where sentient jellyfish invaded Earth and gave us all universal income and healthcare. If physicalism is true, then the limits of what can be are indeed the limits of what is coherently physically realizable (on which see, again, The God Impossible and The Argument from Specified Complexity against Supernaturalism, as well as Defining Naturalism: The Definitive Account). And all it means to say that something is a possible world is that there is a potential physical way to arrange it to exist, that it could be realized, unlike Fermat polygons or square circles or solvable halting problems. Or purely fictional unicorns who can actually stab you. Or presently epistemically probable gods that aren’t evil.
So all actual mathematical and logical truths are accounted for by physicalism.
I believe I actually understood that, which is remarkable in itself. Thank you Dr. Carrier.
Thank you for saying. It’s hard to find ways to make deep subjects easier.
I know, I often struggle with it myself, but you truly handled this topic with masterful clarity.
Entity misguided by the marvel of its own memory—mistaking persistence for truth, and sequence for necessity. If you were a wind, a melon would simply be, and then it wouldn’t, and then it would again. No count, no continuity, no particle to grasp. Just the rhythm of appearance and vanishing, unburdened by names or numbers. Memory carves edges where none exist.
Poetry isn’t really suitable here. See my comments policy for what an appropriate comment might be here.
[Content deleted. AI is against my comments policy. Please never post AI content here. AI is bogus and unreliable. It’s just an autocomplete for the dumbery on the internet and will never provide any insight and is frankly destroying your ability to critically reason out your own thoughts from different qualities of information. — ed.]
If we have models of set theory S1 and S2 say, where S1 allows for infinite sets and S2 doesn’t.
Suppose we can show independently of S1 and S2 that there is some infinite physical quantity. Does that make S2 less valid than S1?
What if S2 allows for some exotic object that S1 does not – is that object real in any sense?
Attend to your use of the word “allows.”
What does that mean ontologically?
If you are restricting what can be in S2, then S2 is describing a different state of affairs than S1, in this case whatever state of affairs would not contain infinities. That would entail an artificial constraint (something you are adding to the system to prevent it realizing certain states), and thus it would not describe all possible things, but just an artificial selection of possible states.
So no, showing that S1 can be realized would not (by itself) entail S2 cannot be realized.
There may still be some other impossibility to S2. For example, there may be no possible way to prevent actual infinities in S2, e.g. if S2 contains any geometric space, it necessarily contains all actual infinities; and thus describing S2 as containing one and not the other would be self-contradictory. And therefore S2 could never be realized: it would be an impossible state of affairs.
As such it could still be studied by paraconsistent logics but would never describe any actual thing except itself, i.e. a piece of paper on which is written all the propositions defining S2 can exist, but those propositions would not describe any other thing than that; only the description is possible.
However, let’s assume for the sake of argument that S2 has been carefully defined in such a way that it can never contain infinities (no manipulation of any elements in S2 would ever realize an infinity). I struggle to comprehend what that would look like. But maybe a mathematician could come up with something (maybe some weird kind of pixelated geometry where geometric points and irrational ratios never occur). But in that unusual circumstance, then S2 lacking infinities does not make its realization impossible (its realization would then be some kind of global “infinity preventer” that locally prevents infinite quantities or ratios forming in any way).
Then, in that extremely weird circumstance, the exotic object of S2 that contradicts S1 (and thus cannot be realized wherever S1 is realized but not S2) would simply be a feature of the S2 machine (that mysterious “infinity preventer” would have some property that cannot be realized except within infinity-prevented subsystems—infinity prevention itself would be the first obvious example of such a property).
However, I think you should not confuse models that “disallow” infinity with models that merely don’t contain it. There being no infinity in S2 is not the same thing as S2 “preventing” infinities. If I have a set of all my clothes, it contains no infinities. But that does not “prevent” there being infinite clothes (even in my wardrobe, much less at all, since whether I have infinite clothes or there are infinite clothes is purely a matter of contingency).
See Mathematics Stack Exchange on this. The negation of the axiom of infinity in ZF results in first-order Peano Arithmetic. But that does not have anything to do with what can exist. It’s just a description of what happens when you make a subset of all possible things that only contains finite quantities of things. That’s simply arbitrary line-drawing. Not a declaration of an impossibility (obviously, as we know first-order Peano Arithmetic does not exhaustively describe all things, not even in actual physics).
Remember the Axiom of Infinity says there is at least one infinite set. The negation of which is not “there is no infinite set” but rather “there is no infinite set within this superset,” i.e. denying the axiom does not entail no system contains an infinite set, only that the selected logic (e.g. 1st Peano) only describes finite sets. Which is not the same thing as saying only finite sets exist.
In other words, ZF -infinity simply demarcates a set of descriptions limited to finite sets. It does not declare ZF +infinity cannot also still exist.
In what sense is restricting S2 artificial? We are free to choose whatever axiom,d we like, from a mathematical point of view we want something that is interesting and a set theory that does not allow infinite sets may still be interesting in the sense that you can still prove theorems from the axiom.
If we say this is artificial can we not also say the axiom of choice is an artificial freedom?
“We are free to choose” = artificial.
You are creating the set. Reality is not delimited to S2. That’s what artificial means.
So, the set of all my clothes is an artificial set. That does not mean my clothes don’t exist. It just means I artificially chose what to put in, and to name, that set. It is not something reality does independently. Reality is not limited to what I put in my wardrobe or you put in S2.
“If we say this is artificial can we not also say the axiom of choice is an artificial freedom?” I don’t know what “artificial freedom” means. But if you mean, the choice to put the axiom into a system (rather than exclude it) is artificial, then yes. Because all mathematical systems are invented (they did not exist before humans made them).
But if you mean, “Is it artificial that a set including that axiom is better at describing reality?” then no. Just as it is not artificial that my clothes exist and so a set including my clothes does describe reality. Even if something described reality as a whole (like the axioms of ZFC—that’s why those are the axioms of ZFC) that fact is not artificial. But the computer language called ZFC is artificial, just as every system of calculus or trig is artificial, but still describes something in reality that is not artificial. That’s why we can have completely different systems of calc and trig: those are made up procedures for calculating something; but the “something” being calculated is (for example) spacetime, a real thing.
So are we saying that S1 and S2 are both valid but only show different “slices ” of a higher mathematics that contains both the infinite sets of S1 and the exotic object of S2, and it’s this higher mathematics that is real in the sense of the article?
What’s to say this higher mathematics is itself not just a slices of something even higher? I suppose, to answer my own question, there would also be a higher reality to go with it.
I think i can accept Cantor type arguments because if someone really wants to argue against it then I could it’s just “abstract nonsense ” and a question of how much abstract nonsense one is willing to allow.
That math is real seems to me to be taking the abstract nonsense and putting it in a real context and maintaining thats it still just abstract nonsense- saying reality is full of abstract nonsense is giving reality a bit too much credit for my liking.
Having said that, if you could establish that even an uncountably infinite quantity of something exists, then I’m would agree everything you say would follow from that
It’s not clear what you mean by “valid.” The only “invalid” sets are sets that cannot exist. But since subsets can exist, no set has to contain everything that exists. Defining a set does not make everything else outside that set cease to exist.
For example, the Cantor cardinalities are each a set. But the existence of one does not forestall the others. The only ones that don’t exist can’t exist, like the set of all irrationals that can be put in one-to-one correspondence with the set of all wholes.
As far as being real, I already showed how all of Cantor’s sets are real, and indeed actually real. But there are obviously many “valid” sets that are real only in the sense of being potentially real, as in, they could be realized even if they never are.
For example, the set of all carboard boxes that test Fermat’s last theorem (every box with three dimensions in ratio to each other for every n) is “larger” than the set of all actual Fermat polygons, because only two Fermat polygons can exist (for n = 1 and 2), and lots of those actually exist. But the “test” set contains infinitely many cardboard boxes (all of which fail to contradict Fermat’s last theorem because none of them have all three edges in whole number ratios to each other because none ever could, e.g. a box of unit size 3^4 + 8^4 = ~ 8.039^4, where the third dimension is not a whole unit quantity, but the set still contains all these Fermat-confirming boxes). We do not need that set to exist to know that it could exist. And that it could exist is simply a physical property of space itself (no Platonic objects or anything else need exist for it to be literally true that a given space could contain all cardboard Fermat boxes).
But all Fermat shapes exist on your fingernail right now. This has been proved by calculus. They are not made of the same thing (most of your fingernail is empty space or EM fields—and possibly all of it, if electrons and quarks just are entangled fields) but there is a traceable (emphasis on -able) Fermat shape anywhere on your fingernail (even if its edges trace only space and not, say, the electrons occupying or zooming through it), and in fact all traceable Fermat shapes are there. Actually. Right now. That set is actually real, not only potentially real (like the cardboard box set of Fermat shapes).
But we know what it would take to create a corresponding space with one cardboard Fermat box for every Fermat shape on your fingernail, one-to-one, and even in the same arrangement. And all it would take is a purely physical arrangement of things. We might have no way of ourselves doing that (we are limited apes) but that we are limited by resources is not relevant to whether that collection of boxes could exist and exist on no other facts but physics (no Platonic objects or gods would have to exist for that infinite box collection to exist).
By valid I mean that my S1 and S2 are axiomatic models of set theory and that, while S1 and S2 are different, at very least their own axiom are not contradictory.
My difficulty is for example, when you say my finger nail contains a copy of continuous 3d space, where is that established and not assumed?
I suppose your fingernail is technically in 3D but I was only referencing two dimensions (e.g. triangles, not cones); adding curvature would make no difference to any point I made, but neither would subtracting it.
So let’s stick with a two dimensional space (the surface of your nail) just to keep it simple: as you look at it right now there are infinitely many geometric points. This is true even if it is not a continuous space, e.g. even if at the quantum scale precise locations become indeterminate and your thumbnail is in some way pixelated as a matter of physics. That makes no difference to the fact that then each pixel still has infinite geometric points. This is a product of the existence of a span.
Assume pixels. There must be a difference between being at one end of a pixel and the other end, or else the pixel would have no length and thus not demarcate a span and thus not be a pixel but a point and we are back to continuous space.
But if there is a difference between being at one end of a pixel and the other end so as to entail a span, there are infinitely many places to be along that span. That anything placed on any one point might in a nanosecond get swept into an untrackable vortex has no relevance to their being a point to land at all the same. There has to be, for there to be a span between the ends of the pixel at all. Likewise even if you can’t crush anything below a pixel in volume, the volume still entails a surface area, which therefore entails infinite geometric points.
So continuous geometric space is a logical necessity of the existence of any span. This should not be confused with continuous physical space. Your computer screen has noncontinuous pixel (hence physical) space but is still continuous space (as the pixels themselves are continuous spaces, and thus sum to a continuous space—if the pixels weren’t continuous spaces, they’d collapse to geometric points, whereby there is no difference being on one side than the other and thus no span between them).
And this has been formally proved: that there are an infinite number of points in any geometric area (just as on any line) follows from the very existence of an area (e.g. for a line this is obvious, and areas are just two intersecting sets of lines).
Hence there is no controversy about the continuity of geometric space. That is not affected by any possible discontinuity of physical space. Because the smallest physical units of space must still encompass a geometric span. If they did not, they’d have no size and thus disappear.
I am comfortable with continuous geometric space but it still seems to me that the argument is lifting abstract continuous space and putting it into a real context and saying all the properties of the abstract space are now actual properties of the real space.
If we take that as being given, then everything else in the article follows almost trivially – i agree that if you allow me one infinite cardinal you have to allow me all of them.
Coincidentally here is a recent video which highlights that some physicists,and even mathematicians do object to the use of infinity
https://youtu.be/4cFgqkXFMEs?si=p4vD9GkRC8QJE-Wo
It’s actual continuous space. There are infinitely many places to be between the left and right edge of your fingernail. Not “in the abstract.” Actually.
That weird things might happen when you land there has no relevance to this point (as those are contingencies of local physics, not a necessary fact of what’s there).
The video you link to isn’t about what we are talking about here. She is talking about the role of using transfinites in physical theories, not the geometric existence of transfinite quantities. Also, that is Sabine Hossenfelder, a known crank who rarely will correctly inform you about anything. She’s like the Deepak Chopra of physics. Physicists generally roll their eyes at her now. I suggest never watching her stuff. It’s disinformation.
Also is there an example of something that has no potential to exist? Not even sure if that makes sense in the context of the article
Tautologically, that which has absolutely no potential to exist is that which cannot be realized, and that which cannot be realized is every self-contradictory system. If there is no coherent way to describe a system, that system can never exist, and thus always has zero potential. Like Fermat Polygons. Everything else is everywhere potentially existent.
There is a subset of potentiality, local potentiality, which has to do with what can be realized out of existing things (a current state of affairs). But that does not exhaust the gamut of all potential things. For example, this entire universe with one single change ten years ago potentially exists at large, but is not a potential outcome here and now, because “here and now” is pathway dependent. But there could be a parallel universe identical to this one with that one single change. That is not logically impossible.
This is a subset of the other because what is logically impossible about pathway defiant potentials is the contradiction between what could have been with what currently is. The only state of affairs that can exist here and now is the state of affairs that actually exists here and now (which we could be wrong about, but that’s an epistemic problem not an ontological one). It would be self-contradictory to say two different worlds exist at the same place and time. Even putting one in a “parallel dimension” to ours is a new location (not here, but “over there,” wherever “there” is) and thus entails no contradiction. That therefore has potential existence. The other arrangement does not.
Like usual Richard you are all over the map on this one and unfortunately mathematics and physics are your weakest arguments (which is saying a lot). Since you link to your own article (https://www.richardcarrier.info/archives/12390) lets start there.
You say in that article:
“The physical universe does not behave anywhere near as simply as our physical laws would have it. We just choose to ignore all the little things that don’t make enough of a difference for us to care about, and thus we reduce everything to a few simple premises like Archimedes did, and from this we get a simple mathematical law. But this is a human invention. It’s an idealization, a fiction of our own devising. It isn’t a complete fiction, as the law will correctly describe most of the physical systems we usually deal with, just not with complete precision, only with enough precision to suit our needs.
This is an important thing to remember, because people like Howell and Steiner constantly overlook it: though we prefer simple and beautiful mathematical theories, reality is always vastly complicated and ugly. Howell, for example, thinks it’s strange that our ideals of mathematical simplicity and beauty help us discover the truth about the world. But in fact, we are using those ideals to construct idealizations, not actual correct descriptions. We make our fictions simple and beautiful, like Archimedes’ Law, and are content with that because it works well enough. But reality doesn’t obey Archimedes’ Law. There is no real system anywhere in this universe that does. Instead, taking into account all those mitigating factors (only some of which I listed), the actual behavior of solid objects immersed in a fluid would only be correctly described by an equation so nightmarishly ugly and complicated, requiring hundreds of inputted variables that we will never actually know the truth of, that it is certainly beyond any human ability ever to construct, much less easily comprehend.
Thus, we use simplicity and beauty as tools to make understanding the world manageable. But the world itself does not conform to them. Hence, for example, Steiner and Howell cannot deduce anything from the premise “the universe is surprisingly simple and beautiful,” because the real world is actually neither. Only our idealizations are both. And idealizations are human inventions.”
Physicists using previously predicted theory supported by math have shown:
https://www.scientificamerican.com/article/physicists-achieve-best-ever-measurement-of-fine-structure-constant/
“After making the corrections, the team derived final measurements during a monthlong run, finally determining the fine-structure constant’s value to a precision of 81 parts per trillion.”
And yes I know you can quibble that this is still an approximation and not simplicity or beauty but you would look ridiculous to quibble about 81 parts per trillion and try to claim this is NOT one to one mapping with reality described by beautiful mathematical equations.
Not to mention your Zeno’s paradox of infinities was already solved by Calculus (try using the limit method). Oh BTW I realize you are trying to use actual infinities but first show me an actual one in reality and then maybe I will buy it because my mind was not blown and to use your terms sounds tin-foil hattery.
You appear to be responding to a different article. You really should post comments on that article under that article.
The only thing that connects here is that calculus solved Zeno, which I myself have said. So I am not sure what you mean by repeating that here.
As for actual infinities, you are staring at one right now: the geometric points on your computer screen are actually infinite.
So, I am showing you an actual one in reality.
And as for that other article, I don’t understand what relevance your point is. That article does not discuss the fine structure constant. Nor does any point you make about that relate to that article or this article. So I’m at a loss what your point is in even mentioning it.
Richard,
Sure I can understand the confusion as I tried to address one portion of your argument so your couldn’t obfuscate the issue as you typically do but I will more fully flush out my point for clarity.
You said in your previous article:
“But in fact, we are using those ideals to construct idealizations, not actual correct descriptions. We make our fictions simple and beautiful, like Archimedes’ Law, and are content with that because it works well enough.”
But here in this article you say All Math is Real. So you want it both ways. I gave you the example of the fine structure constant to refute your point in the previous article that math is a fiction that describes the real world; if there is a near one to one mapping with reality that gives math an 81 parts per trillion basis in reality, calling it a fiction is absurd.
So if your previous article saying math is a fiction is right then this article is wrong. But given I proved your math is fiction argument to be bunk then lets move on to this article to disprove that.
You say here:
“Therefore every real thing in every possible universe is inalienably mathematical. There is no other way to have it exist and not be mathematically describable down to every necessary detail. Which means the converse is then true: any fictional mathematical structure will correspond to a world that can potentially be realized in the same way.”
This is the crux of your argument.
Great I agree that every universe is mathematical but you haven’t remotely proved that every mathematical structure corresponds to a potential universe much less an actual real universe, So you are going full Fractal Mode Mandelbrot sets and you are unanchored from the real world we actually live in.
Also your Geometric points on my computer screen refutation is lame, because if you are right I should paid more for my phone given it has infinite pixel density and resolution. Here is my counter argument to that: There are 100 trillion angels on the head of a pin, therefore God Exists!
There are around 10^80 atoms in the universe so while large is not an actual infinite so you are never going to form an actual infinite (lame geometric points aside). There is no actual or potential physical universe with an actual infinite in them. So go Platonic or go home because physicalism as you describe it doesn’t get it done. This isn’t even touching your Fermat’s Last Theorem massacre explanation you try to give of it. I would say you are a good Sophist but I think that is a oxymoron.
Our simplifications are real. They describe things really happening, to a real approximation.
If what you mean is, “no satellite de-orbits exactly precisely according to Newton’s laws” then you are missing the point of approximating. That results of an approximator will be fuzzy is already understood by the people using the approximator. No one actually thinks any satellite will perfectly down to the nanometer obey the equations. They know the equations only approximate, they do not one-to-one match what happens.
This is why the potential and actual distinction matters. It is physically possible (it is a physically potential) fact that a satellite will fall so perfectly, and thus the meaning of the equations is to describe that potential fact. But that the actual fall will deviate on a bell curve from that is also understood, and is also a true mathematical fact of what is happening physically.
Thus the meaning of all mathematical propositions just is the physical system it describes. When that is good enough for government work, we use the approximator knowing that in physical reality that describes a potential around which physical deviations will occur. It is then describing a potential physical fact that is close enough to the actual physical fact to be useful. No one has any illusions about this. Everyone knows the difference between an approximation (a physical potential) and the reality approximated (a physical actuality).
It’s all physical facts either way. Without remainder. So there is no further ontology required to explain it.
Mathematicians already did that. I am just describing what they did and how it translates to an ontology.
Potentials don’t have to be actuals. So describing a possible physical structure is not “unanchored from the real world we actually live in.” See my example about unicorns.
Unicorns don’t have to exist to be entirely reducible to a physical ontology.
They have to be entirely reducible to a physical ontology to be possible. Not to be actual. Actuality is a contingency of history.
I think you’ve made the lame mistake here. I did not say there were infinite pixels. There are the same finite number you paid for. I specifically distinguished the pixels from the geometric space they span.
This was proved decades ago, when the Axiom of Choice was being vetted. A pixel cannot span a space (and thus even be a pixel) without infinite points in between. Because any declaration of a finite number of points collapses the entire pixel to zero width. The only way for a geometric span to exist is if the space in between is not pixelated. Hence a Planck length would be zero (and thus the entire universe and everything in it would be located in the same place and have a volume and width of zero) unless it spans 1.616 x 10^-35 meters. But to span 1.616 x 10^-35 meters a point at 0 and a point at 1.616 x 10^-35 meters have to be different points, and thus there must be more points in between. If you then take a section (of any size) of that width, the same has to be true. And so on. It can be deductively shown that it can never stop, and therefore for a span of 1.616 x 10^-35 meters to exist, there have to be infinitely many geometric points in between 0 and 1.616 x 10^-35 meters.
I linked to one of these proofs upthread. You can also see some background in this video.
That is not a deductive proof but a random juxtaposition of two propositions.
That a span logically entails infinite points is a deductive proof, a standard finding in mathematics.
We don’t actually know that. You are confusing visible with actual universe. Plenty of cosmologies have the width infinite. But that doesn’t matter. Because the number of atoms is not relevant to anything we are discussing here. That is merely a historical contingency. Not some mathematical necessity or limitation.
There might be. We might even be in one.
But there doesn’t have to be.
As my article repeatedly explains, physical ontology is potential as well as actual. Like unicorns, we don’t need the actual thing to reduce all propositions to a physical ontology. We just by accident happen to have an example of actual (not potential) infinities. As I explain in the article, we don’t need that for my point to hold. We just are lucky enough to have an actual case, even though a potential one would do. So a fortiori no one can claim we don’t have the actual case. And anyone who does claim we don’t doesn’t understand what infinities physically are.
Hi Richard,
I disagree with a lot of what you say in this post. In some cases I disagree because your assertions have not only not been proved, but because there is a huge body of work on the continuum hypothesis that as of the last time I looked, renders it almost unprovable as efforts to TRY to prove it depend sensitively on the axioms you start with (meaning that at best, proof is contingent as is ALL mathematics, so no surprise there). If the CH is true, of course, your stack of power sets is reduced to just two end points — aleph null and aleph prime — as power sets of power sets are just power sets and the infinity of the continuum is the unique cardinality of all continua, unless I misunderstand its statement.,
In others I disagree with them because you seem to be insisting on a view of mathematics that attempts to render it “physical” instead of “abstract”. Much of mathematics is, of course, tied to the physical world, so much so that it is easy to wish to extend “much” to “all”. In one specific sense — the most trivial and useless one — that is even true, as one can’t really THINK about mathematics or REASON in mathematics without symbols, propositions, rules — it is a form of semantics. However, your argument that mathematics IS not only an ontology of sorts with semantic rules that in practice are invented and manipulated by humans and we’re “real and physical” and hence so is mathematics at the very least begs many questions and ignores the real core of the ontological debates.
Ultimately, you are just saying that you’re pretty much a a platonic realist. This is fine with me, but then, so is conceptualism, so is nominalism. Realism can be evident in every single discussion of the color green, because how can we discuss without using the word, and words are real (symbols). Yet “green” remains elusive in the physical world. Try specifying it in the language of physics, and it can’t be done, not really. Give me a “red” light or “blue” light and I’ll give back a frame transformation where the exact same light is now “green”. Worse we can’t even assert that our EXPERIENCE of green is universal, because it is not. I have two sons with red-green colorblindness, and we all live in our personal platonic caves where we can never be certain that the shadows cast on the walls or our cave are the same shadows observed on the walls of the caves we INFER are associated with OTHER shadows on the walls of our caves. Ultimately, “green” is an ongoing experience, not a symbolic entity, and while there is no reasonable doubt that “greenness” is an enormously complex dynamic process happening on real physical wetware in the real world, that’s no excuse for ignoring that this is at best true and useless as far as understanding an ode praising the green of our true love’s eyes is concerned.
Mathematics is all about that — it is the opposite of realism. It doesn’t just explore our existential reality. It doesn’t even explore potential existential realities. Aside from its connection to our wetware and the constraints imposed by using symbolic reasoning to invent, communicate, describe, etc within its semantic rules, in many cases its topics are utterly abstract, not entities that are “real” at all. Power sets as a means of reasoning is a great example — it is difficult to imagine anything more disconnected than developing a successor operation to permit the recursive generation of the counting numbers from minimal principles and teaching a child to count by counting not power sets but their little piggies, including the one that goes wee wee wee all the way home.
Ultimately, the place I think you are going wrong is this. Because mathematics, logic, reason, etc are contingent semantic systems that at best describe closed (or open, via e.g Godel etc) systems of semantic symbols, they have no necessary connection with the real world. Change the axioms, change the conclusions, and axioms are not provable, they are the propositions, premises, assumptions upon which the edifice of theorem and proof are erected. There is hence never a logically necessary connection between existential reality and some particular structure in the dazzling array of structures bored mathematicians have invented to pass the time. The best we can do is empirically establish probable connections between them, ghostly mental correspondences that permit us to reason about the real world and sometimes or even often get it right.
Don’t get me wrong — I think you understand this. The co-evolution of physics and mathematics is pretty much a dance along precisely these lines. But seriously, “infinity” in physics has a very precise meaning — the process of taking a limit in a way that leads to a consistent result, the idea of unboundedness. Mathematicians and set theorists etc often simply short circuit this sort of thing with axioms and abstraction. This is why I think the assertion that “all actual mathematical and logical truths are accounted for by physicalism” is both false and highly misleading. There are no mathematical or logical truths — mathematics and logic together as systems of reason based on unprovable assertions and hence are no more “true” than those assertions. How do you prove an abstract assertion? Are you going to assert that it is “true” that triangles must have angles that sum to pi? Or that any system of mathematics complex enough to represent ordinary arithmetic is “truly” consistent and complete? Has the 200 odd years of mathematics and physics been developed in vain? Mathematics isn’t true. Mathematics is useful. There is no necessary connection between mathematical conclusions and the real world, but empirically we find that mathematical conclusions work damn well to describe the Universe we appear to be living in.
One use of it is to prove that no omniscient God can exist because no sentient being can know its own mind. This is an actual theorem of information theory, contingent on the most modest set of axioms. David Wolpert proved and published it some time ago, but in a context that hid its applicability to the prime assertion of world religions. It can be demonstrated pretty easily by a limiting process — to “know” its own mind, a sentient being has to contain a symbolic representation of both the mind itself and everything that the mind “knows” AND the set of rules — however compact you manage to make them — for interpreting that symbolic representation. The irreducible information required to encode the state the mind itself is never less than the total information content of that mind, leaving no room for the rules. Note that mere equality is not enough — a stone knows the state of a stone by BEING the stone, but it “knows” nothing at all as sentient knowledge and dynamic reasoning are processes involving high level representations.
This is painfully obvious in the empirical observation of the only high-level sentient mind we are aware of — our own. The idea that you could “know” the microstate of everything in your own brain as it is knowing it is patently absurd — we have a hard time holding three or four things in our awareness at once, not the set of numbers — many of them quite possibly REAL numbers in the continuum (see discussion above of CH) that are surely required to represent the brain’s state at any instant, ignoring its manifold connections to everything else in the Universe as an open system! It is also trivially true of the most advanced of the LLM-based AIs we’ve managed to build so far. Their state IS a pile of numbers, and they know nothing at all of them and cannot — note CANNOT, it is IMPOSSIBLE not just difficult — be trained to “memorize” that set of numbers because the set of numbers and its associated reasoning apparatus dynamically implementing them isn’t big enough to hold the set of numbers. Landauer’s theorem — thought involves entropy — is just as true when the entropy in question is INFORMATION entropy as physical entropy.
Anyway, it’s a pleasure to sometimes stop by and read your posts. You are clearly thinking hard and on a good track — I agree with many of your conclusions and much of your reasoning, although I think you are off on a wrong direction about infinity and math-real-world connections. Still, infinity IS the right thing to be off on. Omni-class gods are provably impossible in every useful sense — IF you accept that reason itself applies to the question. Sadly, religious believers do not.
They literally don’t know how. The idea of infinity is a difficult one to grasp even for students of math and science, and information theory is known by name only to a tiny fraction of the world’s population. If you can’t reduce your arguments to something really, really simple, you’ll just preach to this very small choir.
rgb
That has nothing to do with my article.
The continuum hypothesis is a hypothesis about whether a certain quantity can exist. That we don’t know whether it exists is an epistemic problem, not an ontological one. It either exists or it doesn’t. If it doesn’t, it’s for the reasons I explain. And if it does, it’s for the reasons I explain.
So the continuum question has no effect on any point I make here.
To the contrary I reduce abstraction to a physical property. I do not eliminate abstractions. I am a structuralist not an eliminativist. And my conclusion is the same one Aristotle built and the same one modern modal structuralists advocate. There is a huge peer reviewed literature backing me here. So I am not just making this up.
That isn’t what my article is about. It’s true (even Star Wars is “real” insofar as actual movies and books and thoughts about it physically exist; but that does not mean there is a real Yavin in another galaxy with a rebel base on it). But my article is not arguing this. My article is about mathematical and logical truths before humans even existed. So you seem either not to have read the article or skimmed it so recklessly you missed practically every point it makes.
Incorrect. I’m an Aristotelian. Which is an anti-Platonist. As explicitly said in the article I am now suspecting you didn’t read.
Whether my view of abstractions is realist or anti-realist depends on how one defines all those terms.
Traditionally nominalism is an anti-realist position. And this is undeniable in the case of fictionalism, for example; but I’m a structuralist, not a fictionalost, as explained in the article you didn’t read. But I don’t think the demarcation holds up (I think philosophers are misusing these terms).
I allow abstractions to reference potential realities (not just actual ones), which technically aren’t “real” insofar as they do not actually but only potentially exist. Philosophers would thus call this anti-realist. But I anchor potential realities in the properties of actual realities. Which is a real property of actual things.
So really my view is a realist nominalism. I just don’t regard abstractions as “real” in the Platonic sense (there are no abstract “objects”) but I do regard them as real in a physical sense (they describe the actual properties of all real things, since for an actual thing to have a potential is a real property of a real thing, even if the potential itself is never, by happenstance, realized).
So whether you want to call my position realist or anti-realist depends on what you mean by those words. But it is certainly not Platonist. It’s Aristotelian.
And if you define realism regarding x as “propositions about x are true or false independent of human beliefs (or even humans)” then I am a realist. Because “abstract properties” exist (even unrealized abstract properties, i.e. potentialities, exist) as the inalienable properties of all actual particulars, regardless of whether humans know or believe that (or anything at all).
It can modally.
See Mary the Scientist and then Touch, All the Way Down: Qualia as Computational Discrimination.
If Mary has all propositional knowledge, then she has a complete mathematical equation for building a green-experiencing circuit and wiring it into the perceptual circuits of her own brain and thereby experiencing green. Yes, that equation will be as complex as any substantial computer program is today. But it’s still just math all the way down.
I never said they do.
You are confusing necessary connection with sufficient connection.
We make tons of rules systems, for all kinds of games. But the ones that correspond to (and consistently make successful predictions of) reality we choose to call “maths” and “logics.” The rest is Chess or Sudoku or Dungeons & Dragons and so on.
So the connection to reality is not “necessary.” It’s contingent. And that contingency is what makes them useful and thus we elevate them to a status as authoritative. We don’t trust Dungeons & Dragons to predict and describe reality. But we do trust logic and math to. And we deliberately built them to do that, by installing axioms and rules that succeed at that and abandoning axioms and rules that don’t.
So we have intelligently selected this effect. But the question then remains why math and logic never fail at describing reality, even taking into account that we deliberately designed them that way and labeled them as authoritative accordingly. That’s the question my article answers. You do not seem to have any idea what my answer is.
This is a non sequitur.
It’s common for atheists to straw man theism this way. In fact omniscience is properly defined, when steel manned and not straw manned, as knowing all that it is possible to know, not knowing all things tout court. Theists are well aware God’s omnipotence does not extend to doing logically impossible things, like obtaining logically impossible knowledge.
But though you can’t conclusively disprove God this way, you can reduce confidence in the belief that God exists by pointing out that we can have knowledge God cannot, yet which is crucial to making moral decisions about us. Which kind of ruins the whole point of God, even if, technically, this hamstrung God could yet exist (though by another argument it can be suggested, he cannot).
But that is all off topic here. I suggest any further discussion of thus side point be continued in comments on those other articles I just linked to in that last paragraph.
An omniscient God could not know many things, but one thing they would know for certain is that they are not, actually omniscient. You already brought up star wars or middle Earth. No logical or observational argument can rule out the mathematical possibility that entirely disjoint space-time continua exist where those stories in our world are literal truth. God would know that its knowledge could at best extend over what it could “see” and might well be incomplete. Which would make God no more certain of the true state of all things than we are — limited by our range of metaphorical vision — and would mean that God would be lying if it asserted that it knew all things. And among the many things it didn’t know would be the state of its own mind.
In other words, God could act, but not even God could know, completely, why it chose to act. “Because” is an answer that might be satisfying to three year olds (or their parents) but it won’t do for God. This is a serious objection, in other words. Read Wolpert’s paper on the impossibility of Laplace’s Demon. No computational device within a universe can perfectly predict every aspect of that same universe. Since the Universe (defined to be everything that actually exists, that is the objective standard by which being/truth/knowledge are measured) is a superset of God (if God exists and is a “computational device” capable of thinking at all) then God cannot know/predict every aspect of the Universe. Asserting that omnipotence doesn’t enable God to perform contradictions is irrelevant. We’re not talking about creating a weight too heavy to lift. We’re talking information theory and the fact that no thinking being INCLUDING God, if God exists, can have zero information entropy in the real world Universe of real-world information.
Also, your article above references “infinity” many times, so I’m not sure I see how my pointing out that you are making unfounded, unprovable assertions about infinity are irrelevant and not how the term is used in the most concrete of scientific disciplines, physics, includig poetry about infinities in a fingernail. Prove them! Without making assumptions that cannot be empirically validated and do not depend on a number of assertions that in fact, might or might not be true! But it’s your blog, so I’ll retire on this note as you wish.
You’re playing word games again. As I said, omniscience means knowing all that can be known, not knowing what can’t be known. So it is absolutely possible to trick a being into honestly and rationally believing it is omniscient in that real sense. Hence my point.
Everything I said corresponds to all the peer reviewed literature on transfinite geometries and quantities. And it all has been proved (since Cantor and beyond). So it would seem you are the one not up on the literature or the state of the field.
By the way, one comment on a reply you made and assertion you made in the article is clearly incorrect. You assert that I am seeing an infinity when I look at the screen. You can be forgiven for inferring that, but empirically I’m doing no such thing. For one thing, the information carrying capacity of my retina is very, very finite. So finite that a company just demonstrated that they can build electronic transducers with the same photoreceptive bandwidth as the retina (similar discretized resolution). The same is true for our ears — especially as we get old. Finite numbers of hair cells in the human cochlea, with finite resolution and old age taking its toll. There are finite numbers of nerves leading into the brain, finite numbers of axons and connections, even finite numbers of states available to all of the microscopic moving parts that make UP axons and neurotransmitters. Absolutely nothing in the process of cognition and perception has infinite bandwidth, and most of the available channels for information to enter “us” — our sentient/conscious perceptions — are noisy as hell, easily corrupted, often broken, and subject to weird forms of internal feedback such as when we “imagine” things. Our transformation of the data stream into immediate, short, and long term memory is even more entropy prone — only a tiny, tiny fraction of what we see at any given instant is stored long enough to count as actual cognitive “seeing”.
So no, neither you, nor I, nor anyone on this thread has seen an infinity. In order to do so in any useful sense, we’d have to have infinite bandwidth in our sensory channels and infinite information storing capacity, and we have neither. What we do — and again, it is quite reasonable but is not the same thing as seeing is infer infinity by extrapolation and interpolation on the basis of a mental model. You are literally asserting the the gap between what you actually perceive and reality is interpolatable. But this process is precisely isomorphic to the process of seeing wooly little lambs and sheep in the clouds, or seeing a bunch of dots on a graph and saying “behold, this smooth lie I draw that sort of interpolates the data is reality, and the dots are reality plus some unknown source of noise” (which sounds better but is just as mistaken except when it work well enough over a very long time to the point where we come to sort-of believe it is likely to be true).
Interpolation, extrapolation, are all ways of making numerous unprovable assertions about the nature of being and taking limits. Infinity in the real world is a matter of our overactive imaginations conceiving it and proving that the idea can be made meaningful and/or consistent in the imaginary domain of mathematics, and then applying the concept to our finite perceptions in an attempt to extend them beyond what we actually perceived. Ask a physicist “is the Universe infinite” and honest ones will reply “we don’t know”. SERIOUSLY honest ones will reply “we cannot ever really know”. Even if we found a model that worked nearly perfectly to describe the post-big-bang Cosmos we can see and required the Universe to be infinite would not be proof that the Universe WAS infinite. And no, we can’t get down to the Planck scale in the other direction either. Good physicists don’t even “believe” in magnetic monopoles even though they have enormous explanatory power, simply because no experimentalist has managed to put reproducible salt on the tail of one in the wild.
I repeat: Infinity is an abstraction, a one word compression of the idea that since the successor operation has no upper bound in the abstract world of axiomatic number theory “infinity exists” in the abstract world of axiomatic number theory and hence may exist — or be useful — in our efforts to generate useful knowledge of our very definitely finite perceptual reality via extrapolation and limit taking. Nowhere can we point to something and say “that which I point to is most definitely infinite”. Nowhere can we point to something and say “that which I point to is the pure embodiment of 2 + 2 = 4”, either, but we’ll leave that for another day. The reason it fails is because even if you are adding up piggies, every piggy we can point to is unique. In space and time, even — assuming that the piggy now is the same piggy that it was five seconds ago contains that dratted word “assume”, meaning that it is not known beyond any possible doubt. Identifying these unique space-time event collections with the single tag “littlest piggy” involves an enormous amount of entropy. Language and thought itself are all about information compression, input from an enormous Universe with nearly all of the detail erased as irrelevant, digesting it to where our comparatively tiny minds can do something with what is left.
rgb
You are confusing counting with being. You don’t have to count the melons for there to be thirty of them. So what your retina and brain are calculating is not relevant to my point. You can’t say thirty melons don’t exist “simply because” you can’t count them, your brain just sees a vague “lotta melons.” The thirty melons still exist. And you are still looking at them. Ditto an infinity of geometric points.
Which I explained. Hence, again, it looks like you didn’t read my article, and when I caught you not reading it, you randomly went back and for the first time read something in it, out of context and carelessly, and thus complained of something I didn’t say but even explicitly said I wasn’t saying.
So I think there is something wrong with how you are comporting yourself here. You are not engaging with any of my content seriously.
I’m reading “Negative Math” by Professor Alberto Martinez – excellent. I’m also a big fan of Alain Connes (Fields Medal) and math prof. Louis Kauffman. But I defer to physicist Eddie Oshins and physicist Basil J. Hiley (both of them gone). John G. Williamson and Martin van der Mark – more physicists now passed. My own quantum physics professor Herbert J. Bernstein was also Lee Smolin’s first physics professor in quantum physics.
I am uncertain what degree of math and physics Dr. Carrier is educated in. From what I understand, his doctorate and all preceding education was in Ancient History. I think that possibly it would be better for the doctor to stick to his expertise when it comes to history and philosophy rather than to try and make extrapolations in other fields. Given his education, I feel like it also might be in his interest (if he would prefer to make assertions using mathematics and the natural sciences) to engage with a peer with an expertise in the field to assist in such writings. Yet, I understand that sending a blog post to a mathemetician acquaintance may seem a bit laborous.
I also understand this is more seeking to explain an ontology to those unstudied in STEM than to convince any mathemeticians, as much of it is explaining examples of Freshman or highschool level Calculus, so possibly I’m being too strict.
My mathematics qualifications are summarized in another comment thread. I actually have several peer-reviewed mathematical publications and about a year of diverse college credits in the subject.
But more importantly, a fallacy of argument to authority is evasion, not engagement. If there is an error in this article, point it out. Otherwise, why dismiss it? If all its premises are true and all its conclusions follow without fallacy from them, then you should heed the results no matter who produced them. Right?