“Continuous” circles, at least.

The circular objects we encounter everywhere are discrete entities with jagged edges. They are “imperfect” from the perspective of a lofty mathematician, but for our humble purposes, they are actual circles—one of these grotesque things:

Since space is everywhere fundamentally discrete, all geometric objects are existing within non-Euclidean space. That makes their properties not easily captured by high school mathematics.

I just want to focus on one idea in discrete geometry: what length is, and by extension what a circumference is, as finitists who reject the notion of “continuity” and “real numbers.”

The first thing to understand is what we’re giving up, by rejecting continuity in geometry. Take this wonderful, intuitive, GIF that demonstrates the relationship between diameter, circumference, and pi.

The idea of circumference is that you can “unroll” the “length” of the boundary of the circle. The distance traveled by a rotation of a circular wheel is fundamentally connected to the diameter of the wheel. If the diameter of the wheel is 1, then its circumference is pi—3.14159…

This is all simple, beautiful, and wrong. The GIF is lying.

The first thing to note is that the circumference is fully and completely unrolled on the number line—there is no transcendental pi here. There are a finite number of pixels composing the edge of the wheel and a finite number of pixels composing the “unrolled line”. The ratio of circumference-pixels to diameter-pixels can be expressed by a rational number.

Triangles First

At the surface level, this pixelated geometry approach seems to break our math. For example, if the hypotenuse of a right triangle is simply “the number of pixels which compose it”, then the standard expression of the Pythagorean theorem does not hold—depending on how the lines of the triangle are constructed, c might be equivalent to b (i.e. the hypotenuse and its base have “the same length”), or in the stair-stepper pattern, it might be exactly 2b.

The problem with triangles it the same as the problem with circles. Let me attempt two ways to explain:

Attempt 1

The problem is treating length as dimension-independent—treating “length along A, B, and C” as interchangeable lengths. But if you look closely, there’s something unique about length C which is totally different from lengths A and B.

Think about it in terms of difference. The bottom of Line A is different from the top of Line A along one dimension. The leftmost point of Line B is different from the rightmost point along one dimension. But line C combines these two changes—it’s different along 2 dimensions!

So my claim is that talking about “length” means something different when there is dimensionality (or shape) involved.

The assumption of continuous space discards this information and says A, B, and C’s length-units are all interchangeable. In other words, if A is 3 feet and B is 4 feet, then C is 5 feet. These feet are all interchangeable and have the same meaning—feet are feet!

Attempt 2: Mathspeak

I’m sure there’s a way to encapsulate this idea with mathematical language, but I don’t know enough to be confident. From what I can tell, this is essentially about the question of isotropy versus anisotropy—whether or not a space is uniform in every direction.

1d discrete space might be isotropic, but as soon as we move into two dimensions, it becomes inherently anisotropic, and with anisotropic spaces, we cannot represent length with a simple scalar value. We need higher-dimensional language.

My claim is that isotropy is a mathematical fiction; all isotropic spaces bake in the assumption of continuity. Without continuity, all space is anisotropic.

i.e. real space is discrete space, and discrete space is anisotropic space.

Back to Circles

Returning to our wonderful GIF:

I want to say it’s continuity that is creating all the problems here.

In the standard story, we would say something like “with a perfect circle, the ratio of its circumference to its diameter is irrational, and therefore at a deep level, the notion of ‘length of the diameter’ and ‘length of the circumference’ are incommensurable.”

However, that story does not hold if circles are discrete. By rejecting continuity, the irrationality and incommensurability go away too!

So, there should be a way to meaningfully talk about the ratio of a circle’s circumference to its diameter with total precision—without invoking irrational, real, or transcendental numbers. I don’t yet have the mathematical language to express this, but I know it will look different than simple scalars.

All of this, of course, vindicates one of the criticisms we finitists receive: we have taken a very simple thing and turned it into an incredibly complex thing. But alas, that’s the cost of moving from “good enough, useful, approximate models” to “total logical precision.”

Norman Wildberger might be the guy to place mathematics on firm foundations.

In the story I’m telling about history, our present dark age was precipitated by the foundations of mathematics being undermined in the late 19th century. The castle was destroyed and never rebuilt.

Wildberger is trying to rebuild the castle, finally.

Despite my criticism of set theory, I have actually become persuaded that set theory is indeed the foundation for math — but finite sets, not infinite sets!

The fact that his upcoming book is titled: "Box Arithmetic: A Finite Multiset Foundation for Mathematics" made my spine tingle.

It’s way, way, way too early to say whether he’ll accomplish the mission. But he’s the only person I’m aware of with technical sophistication who has possibly found the right foundations.

I felt excited and obligated to tell you, dear reader, the day his video came out :).

(For what it’s worth, if this does check out, I will likely be spending a large amount of energy, and part of my career, fleshing out the implications of this work.)

The differences between mathematics, physics, and logic.

The difference between true mathematics and speculative mathematics.

Mathematics is a language, and like natural language, it can encode real patterns that describe reality. However, also just like natural language, mathematics also contains a bunch of speculative, paradoxical bullshit. I am deeply suspicious of all mathematical “models” of the world that try to predict behavior (i.e. mathematical physics); they can be useful, but it’s ultrahubris to think we’ve discovered any final laws of physics. All of them, in my mind, are up for grabs -- literally all of them, including conservation laws, etc.

Logic on the other hand is, in some sense, part of the operating system of reality, but that’s saying less than it might initially seem. That just means “the system is however it is; it operates however it operates” which isn’t even a “constraint” per se, because it couldn’t even be otherwise -- unlike any law of physics, which we can imagine revising without contradiction.

The true parts of mathematics are built on logic, as an extension of logic -- so arithmetic can be understood as the logic of quantity. 2 + 2 = 4 is going to be true always and forever, but it’s not really a “model” of the world; it’s just a true way of describing a quantitative relation. Since reality contains multiplicity (i.e. there is more than one thing), arithmetic can tell us some true aspects of reality.

I’ve been saying for many years that we’ve been living in a dark age since at least the early 20th century -- lots of people seem to resonate with that idea. But then I’ve also been saying: and the philosophy of mathematics is the domain in most need of revision, and people are much less excited about that. [Bob] is one of the few people I’ve spoken with who seems to have read and thought enough about this subject to see why that’s a plausible position, because we really went through massive revolutions at the foundations of our thought between ~1870-1945.

In discovering wonderful and important things about math and its relation to the universe (e.g. Euclidean geometry is not necessarily true), we wound up throwing out the baby with the bathwater -- that is, logic.

Simon Dixon and I talk about theories surrounding the hijacking of Bitcoin, the connection with Jeffrey Epstein, and some history about Bitcoin's funding and development.

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I am not saying this argument is true, merely that it’s reasonable, coherent, and understandable in the context of standard philosophy.

Here’s the argument:

1) Patterns are real and mind-independent (we’ll call them “Forms”).
2) There is a metaphysical hierarchy of Forms.
3) At the top of the hierarchy is the Form of the Good.
4) The highest form is Love.
5) Therefore, Love is the Form of the Good.

We can break this argument into two parts: Propositions 1-3, which can be understood as a simple Platonism (using the word “patterns” instead of the usual term “universals”); and Propositions 4-5 which can be understood as an empirical Christian claim which clarifies and updates the Platonic idea.

What I mean by Platonism

By “Platonism,” I do not mean a faithful reconstruction of Plato’s ideas—rather, a metaphysical framework which says that patterns, abstractions and relations are real features of existence, separate from our minds. There are cats, and there is Cat-ness, the pattern which all cats have in common, which sits outside the material world in an abstract domain.

The point of this article is not to make an argument for this kind of Platonism (which I’ve done extensively elsewhere.) The point is to say that within this view, there is a hierarchy of Forms. As you go up the abstraction ladder, you reach higher and higher forms. Eventually you’ll find the Form of Justice Itself, which is an extremely high-level, high-dimensional pattern that every just action participates in.

Impersonal Form of the Good

At the top of this hierarchy, the Platonist wants to say, is the Form of the Good. This form is so absolute, so fundamental, that it’s the Form which grounds all of our ethical judgements. The form of the Good isn’t something which can be judged “good” or “bad”—it is Goodness itself, the Form-that-we-use to determine goodness.

Functionally, this Form plays a similar role to God in theism. However, the classical Form of the Good is critically impersonal. If there is a Creator-God (or “demiurge” in the Platonic vocabulary), he sits underneath the Form of the Good, participating in it. The theistic God, by contrast, has nothing above him; he sits at the apex of the metaphysical hierarchy.

That gives us the meaning of Propositions 1-3 of the original argument. Now the Christians come along.

Empirical Revelation

The Christian wants to say: by the way, it turns out the Form of the Good is not a distant, impersonal abstraction.

The Form of the Good turns out to be Love.

In other words, Love is The Good. Love is Goodness itself—it’s the thing which we use to determine the goodness of everything else; it’s the Value underpinning all other values; it’s the objective meaning of life; it’s the first and final cause of everything.

And Love is in the Form of a Person. Love is alive.

This is not an aprioristically-true claim. It’s an empirical claim—that is how we can understand the meaning of “divine revelation”—God reveals his face, not through a purely logical argument, but through Incarnation.

The Form of the Good “comes down from [Platonic] heaven” to live among us.

Immediately, if such a thing happened—or if it plausibly happened—we would find ourselves grappling with the same philosophical questions as the early Christians. Divine Love is fully instantiated by Jesus Christ. So, what was Jesus’s exact relationship with God? Clearly he was a man, yet divine at the same time... and suddenly we’re back in the 4th century.

So that’s the claim. If there is a hierarchy of Forms, then the Christians are saying Love is the highest Form and the essence of God.

Note:

If we want to replace the Platonic conception of the “Form of the Good” with the Neo-Platonic idea of “the One,” the Christian argument still holds.

So we can swap out premise (3) for the claim “At the top of the hierarchy is The One,” which would change the conclusion to: (5) Therefore, Love is The One. Either formulation works.

Part 2:

Patri’s website: https://patrifriedman.com/

And yes, this narration was generated by an AI clone of my voice. Incredible.

Free State Project: fsp.org

Despite being a common fallacy among the scientific class, it is inherently anti-scientific. It causes people to dismiss observations out-of-hand without examination due to discordance with existing theoretical models—that is, anti-empirical in a technical sense. It also tends to reflect a lack of creative thinking. Generally speaking, we see connections before we understand them—sometimes these connections are observed thousands of years before we can pretend to understand them. People have been using soap and boiling water to stay healthy long before anyone could explain why.

To put it bluntly: we don’t understand causal mechanisms. Even for simple things. We tell ourselves nice-sounding theoretical stories, at a certain level of resolution, that usually turn out to be false and sloppy. Our resolution is not fine-grained enough to truly understand almost anything. If we required mechanism to be sorted out before entertaining hypotheses, we would not be able to offer any hypotheses at all.

Simple and Common Medicine

Let’s start with a concrete example: aspirin. What could be a more commonplace medicine, prescribed for ages, for all kinds of ailments? Not a fancy medication at all. We know all kinds of things about aspirin — its chemical composition, generally safe doses, generally safe contexts for ingestion—and yet it turns out in 2025, we still do not fully understand its mechanism of action (Example 1, Example 2, Example 3). We can use it, but we don’t fully understand it. There are stories we tell about mechanisms, with progress made every few decades, but nothing is finally figured out. To make the point explicit: if aspirin were a new medication, it would thus be self-evidently wrong to say, “We should not believe in its effectiveness because… what’s the mechanism?” Clearly, knowing that aspirin works is a completely separate question from understanding why it works.

It will probably take a much fuller (perhaps complete) understanding of the human body to really figure out how simple aspirin works. And alas, it will probably take a full understanding of physics to figure out how the human body works.

Balls and Needles

How about an ultra-simple example. A ball rolling down a plank. Surely we understand what causes this basic phenomenon. It’s gravity, of course!

I’m afraid it’s not just gravity. It involves a list of concepts so large, I can’t even list them all—surface tension and air resistance, thermodynamics, geometry, time, quantum physics, each one of these domains splinters into a million more complex sub-domains. And come to think of it, we don’t even have gravity itself figured out. Whose gravity—Einstein’s or Newton’s? To truly understand the ball rolling down a plank, we probably need a full understanding of quantum gravity, which does not yet exist. So unless you are willing to claim to have solved physics, it’s fair to say we don’t actually understand the true mechanisms for utterly basic phenomena. We tell nice-sounding stories that answer surface-level questions, but our stories are certainly incorrect and incomplete. Therefore, when people make claims about unusual physical phenomena, we should be extremely careful before dismissing those claims due to a lack of known mechanism.

Take another example: acupuncture. Much has been written about the effects of acupuncture and the usual woo-woo theories accompanying it. There are legions of scientific minds who dismiss the practice due to a flimsy theory of mechanism—energy flow, meridians, qi, etc. And yet, despite the mockery from scientific Westerners, the practice continues to grow in popularity all over the world.

The trouble is: there’s a ton of evidence that acupuncture works, both in the medical literature and outside it (Example 1, Example 2, Example 3,). We just don’t understand the mechanism yet, though there are several obvious leads. The nerves in our bodies, for example, match the “energy flow” diagrams with fair accuracy. It’s not hard to imagine that the tiny needles are interacting with our nervous system in ways we don’t understand. In fact, I’m sure in hindsight that will be utterly obvious.

Prayer and Needles

Two more controversial examples: prayer, and the “safe and effective” medical intervention that people have strong opinions about.

Prayer is an easy one, and it doesn’t take too much creativity to come up with a plausible mechanism. At the very least, praying obviously benefits the pray-er, in the same way that meditation benefits the meditator. If this is true, then it’s not hard to speculate how prayer might help others too, without appealing to any mystical mechanism.

When somebody says, “I am praying for you,” they are saying “I am caring for you and sympathizing with your situation.” That feels good; it is going to have a non-zero effect on your nervous system (as receiving a hug and a kiss might, too). When fifty friends are sending somebody their prayers, the pray-ee usually feels good about it. They might even describe it as “feeling the love” which is going to concretely affect their nervous system. Not to mention the widespread social benefits accrued to members of the group participating in the communal prayer act (and the effects on their neurology and group-bonding, which provide even more routes to benefit the pray-ee).

Furthermore, we know there’s a connection between brains and electricity. Brain waves are a real, measurable thing. We know meditators can affect their own brain waves—and by extension, since those waves stretch through the skull, they have a non-zero impact on the environment around them. This is why medical apparatuses can “read” brain signals—because those brain waves actually physically affect the world. It’s not difficult to imagine that prayer (especially coming from a skilled prayer/meditator or many people) might have real physical effects in the complex domain of bioelectricity.

This is just simple, easy, obvious speculation. What are the chances that the practice of prayer, discovered since time immemorial, therefore is tracking something real, even if we don’t understand it? I’m going to say it’s practically guaranteed, especially if we account for indirect mechanisms of interaction, much less any supernatural causality.

The last example is vaccination. On the one hand, we have an army of mothers around the world claiming they have observed a causal connection: shortly after receiving a shot, their kids started to develop neurological problems. On the other hand, we have an army of medical professionals insisting the science is settled, that there’s no possible causal connection, etc.

In this case, there are several plausible mechanisms for neurological damage, but even if there weren’t, the empirically-minded scientist must recognize that if the signal does not go away—if mothers around the world are still making the same claims after decades—then that’s evidence of an unknown causal connection. That’s exactly what it looks like when there’s correlation with not-yet-understood causation.

Anti-Empiricism

It should be noted the irony of the extremely anti-empirical approach of a mechanism-first understanding of nature. Empirical inquiry starts with observation. To reject the validity of observations because they do not fit pre-existing theoretical models is perhaps the definition of dogmatic, anti-empirical thinking. Unfortunately, it’s made legions of 19^th and 20^th century thinkers extraordinarily arrogant and has blinded them to basic observation. They are convinced their theories are so good, there’s no longer a need to open their eyes and observe the world!

As a general rule, I propose that the question “But what’s the mechanism?” should never be considered a counter-argument to any claim. It can be asked in good faith to prompt people to start thinking creatively, but it’s not criticism, especially considering our shallow understanding of the world. There’s a high likelihood all our favorite scientific ideas will be invalidated in the next few centuries, so let’s think with a little more humility.

Guest's website: https://hope.edu/directory/people/ortiz-jared/index.html

Guest's book: The Nicene Creed: A Scriptural, Historical, and Theological Commentary

My conversation with Michael Button, a rising star from the next generation of scholars. Check out his YouTube channel here.