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?

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.

“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.

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.

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

Mathematicians already did that. I am just describing what they did and how it translates to an ontology.

So you are going full Fractal Mode Mandelbrot sets and you are unanchored from the real world we actually live in.

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.

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.

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.

Here is my counter argument to that: There are 100 trillion angels on the head of a pin, therefore God Exists!

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.

There are around 10^80 atoms in the universe

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 is no actual or potential physical universe with an actual infinite in them.

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.

abstract continuous space

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.

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.

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

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.

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?

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).