Through the Heart of Every Man

An Unpleasant Surprise

1 February 2024 | Libertarianism Philosophy Philosophy of Mathematics

I'm a libertarian, interested primarily in mathematics, physics, computer science, and economics. I have a recently-acquired pair of degrees in the first two, feel better-read in their foundations than most of my peers, and readily admit the existence of major foundational problems. I myself believe the academy to be irreparably corrupted, which is why I abandoned my former plans of a physics Ph.D. to work in industry for a while, and was going to attempt to lay out a model for meeting the demand for knowledge without force.

So, that explains my excitement on hearing Steve Patterson's planning to do just that on Tom Woods' show today. I followed the links to his socials, interested to jump on board with this new movement. However, what is offered might well worse than the disease: after reading his quarrels with prevalent philosophy of mathematics, I find a downright muddy thinker that hasn't even bothered to consider the beliefs he attacks on their proponents' terms. I find someone supporting a logical view of mathematical foundations who apparently hasn't even bothered to learn what "logic" or "a contradiction" is except presumably through the leaded-glass distortions of a novelist-wannabe-thinker.

This has severe consequences for what he is attempting to build: by having a downright terrible understanding of the objects of his critiques, he will be eternally shadow-boxing, ignoring the actual challenges that lead to terrible predictions and sordid manipulations that have come to the political forefront recently.

Humorously, Patterson brings up Gell-Mann amnesia. For me to trust that his critiques of other disciplines are accurate would be nothing but a further instance of that phenomenon he criticizes. A line-by-line refutation of his article "Infinite Things Do Not Exist" will illustrate what I mean (only the parts with some substance included).

First of all, we have to define our terms. "Infinity" or "infinite" means "without end," "never-completed," or "without boundaries."

Indeed, first of all we do have to define our terms. If we are trying to refute foundations of mathematics, however, we should actually use the definitions from foundations of mathematics, so our arguments actually have teeth.

Infinity, in foundations of mathematics, only applies to sets, or unordered collections. These are the only "things" which comprise the ontology of mathematics; the rest of the taxonomy of mathematical objects are built from or coaxed out of them. A refutation of the general metaphysical concept of infinity, as he has attempted to construct, therefore bears no necessary relation to the term-of-art "infinite" in the context of this ontology.

Indeed, the mathematical definition is actually meaningfully different: they claim ∃X[∃e(∀z¬(z∈e) ∧e∈ X)∧∀y(y∈X → y∪{y} ∈ X)] (Zermelo-Fraenkel axiom of infinity, cf. Wikipedia). I.e., there exists a set containing the empty set and all sets buildable by "iterated nesting" starting with the empty set, i.e. there exists a set containing all von Neumann ordinal numbers.

The claim "Y is infinite" means only that there exists a surjective map from Y to X, i.e. one can formally define a correspondence between elements of Y and elements of X that doesn't "miss" any elements of X. There is no concept of "without end," "never-completed," or "without boundaries" here without adding more exogeneous structure that allows one to talk about "end," "completion," or "boundary."

An infinite distance can never be covered—by definition of what we mean by "infinite." There is no "end" to an infinite series—if the series ends, it's finite by definition

Indeed, these are all true statements, but not sound statements—they do not follow from either the definition of infinity he's presented, nor from the definition of infinity he should be refuting. They require a definition of the concept of "distance" in such a way that infinitude applies to it, a corresponding definition of what it means to "cover" something, a definition of "end," a definition of "infinite series," and the introduction of the (comparatively less leap-laden) notation "finite" for the first-order formula "is not infinite."

Consider the question, "How many positive integers are there?"

Most people intuitively answer, "There are infinitely many positive integers." Meaning, there isn't some upper-limit on the size of numbers. You can't think of a number that "1" cannot be added to. This conveys the general concept of "infinite."

I don't know how what "most people intuitively answer" in any way influences the precise sense of the term used by experts, any more than what most people mean by "value" or "prefer" influences how Austrian economists are permitted to use it in a technical sense. Certainly, what he argue against cannot be the technical concept if he defines terms as above.

By the term "actual," I mean "fully-realized," "completed," or "totally encapsulated."

I don't begrudge anyone their definitions, but this is literally just assuming his conclusion, and is totally divorced from any use of the term "actual object" by any person anywhere in history. Additionally, the words he equates "actual" to are inquivalent and have no obvious commonality that I can extract and assign to the new referent he's assigning to the word. By its standard, no emergent objects exist—a waterfall is a fundamentally dynamic entity (its constituent parts are continually changing), and so the waterfall is never fully-realized or completed, because its essential nature is one of change. It is, however, totally (spatially) encapsulated, which illustrates the differences in the "synonyms" he attempts to use here.

And here we find the elementary error in the conception of an "actual infinite". (sic) I realize my refutation would appear impressive and profound were it complex.

It would appear impressive and profound were it:

1. actually addressing the belief he claims to refute,
2. using "actual" in any sense other than one he made up for the sole purpose of being able to refute the nonexistent belief he's outlined,
3. actually to contain a valid instantiation of the concept of reasoning it is asserted to employ.

If it were some difficult, abstract chain of reasoning disproving a century of mathematical thinking—that would surely impress people. But alas, the refutation is not complex. It's outrageously simple. So simple, it is anti-climactic.

What is never-completed is never completed.

This is simply a false implicit claim about what counts in mathematical or philosophical practice. Look at the corresponding sections on ViXrA, and you'll see any number of difficult, abstract chains of reasoning that nevertheless are considered so crankish and obviously errant as to get their authors booted off the extremely lightly moderated ArXiV from which the service has forked. Difficult, abstract chains of reasoning are incidental instruments to the objectives of the disciplines, not their raisôn d'etré. In fact, simple, crisp, elegant (correct) arguments are considered the crown jewel of these domains.

Moreover, any time you have any philosophical argument on an issue where there exists substantial disagreement, the barest breath of intellectual humility should cause one to be suspect when it is "too cheap." This arrogance seems to be an all-too-common common vice of the liberty-oriented; I've seen it happen over and over again with midwits on the internet thinking Hoppe's argumentation ethics provides unassailable proof of libertarian ethics despite their not adopting any of the epistemological premises that are essential to the reasoning.

After this point, he engages in even worse sophistry. I've refuted some highlights.

What is the curvature of a circle with an infinite radius?

Never in this entire section does he define what "curvature" or "circle" mean, and never gives any reason why the two must necessarily connected. That he doesn't even take the time to consider his implicit priors here is simply baffling to me, and it communicates no actual on-the-ground engagement with mathematics or logic.

Believe it or not, some mathematicians will say "That is not a contradiction! This just shows the incredible nature of infinities! Paradoxes exist, you've just proved it!"

He gives nothing approximating an argument, much less a proof. The extent of his reasoning is "try to imagine," and then giving a description of what he imagines. There's nothing presented, at all, that would constrain what the reader has imagined. Because, again, he doesn't define what is actually meant by "circle."

This example has very little to do with any "incredible" nature of infinities¹, and everything to do with the definition of the figures in question, and the limiting process involved. I would advise reading the most minute amount on projective geometry to clarify this example.

Though calculus can easily be rescued from logical contradictions, set theory cannot. The set theoreticians are absolutely explicit: according to them, some infinities can be fully completed; they have an actual size.

This paragraph almost had me throwing my phone. Not even actual finitists (who do exist, albeit in small numbers, in academia across the globe) are this dishonest about the premises they dislike (they're mostly imanent realists).

First of all, the axiom of infinity is one you can easily drop, and do finite set theory. This is "rescuing" set theory from "logical contradictions," and I'm sure it's much easier than whatever tortured interpretation of calculus he's come up with. Second of all, he's clearly had no engagement with what set theoreticians are actually saying: they have a specific definition of "size" called "cardinality," and confusing it for other characterizations of size that coincide on fintie sets are where most "paradoxes" about infinite things arise. If he had actually engaged with set theory, he would both:

1. know that cardinality has a specific meaning distinct from colloquial size, and
2. that it is defined literally just by taking a representative of an equivalence class under bijection, i.e. the "size" of a countably infinite set is by definition just the natural number set (just like the "size" of a set of 3 objects is the number 3, which is by definition a particular set with 3 objects). This does not even implicitly or abstractly carry with it any notion of "completing" an infinite process.

Now re-examine the concept of infinity. "Infinite" means "never-ending," "incomplete," "always-bigger-than."

But "always-bigger-than"is another way of saying "Not merely A. More than A."

In other words, the very term "infinite" is an explicit denial of identity

Therefore, an "infinite thing" is "a thing which is itself, and more-than-itself at the same time." An outright contradiction.

This passage is where I stopped reading, as it became clear that he has not even had any interaction with logic, philosophical or otherwise. Because this simply is not what "contradiction" means.

First off, he's done the thing again where he's stated three inequivalent words as if they were synonyms in order to define a concept in a way that is entirely private so that he can delude himself into believing he's made an argument. An infinite set is not (strictly) bigger than itself. In the case of the integers, its elements always have larger elements. These are completely different orders. These are different classes of objects. I can't possibly imagine how anyone could make this most basic of object-type errors and expect to be taken seriously in their opinions on philosophy of mathematics, much less found an institute on the basis of said errors. An infinite set, indeed, is what it is. It simply is not more than itself; no one since Cantor other than he has ever attempted to abuse concepts to assert so.

Moreover, the final nail in the coffin, in order to assert that something is a contradiction, he must demonstrate that the premises entail both A and ¬A. He does not even attempt to give an argument why "is" and "more than" are such that "thing" and "more than thing" are necesarily different. He merely asserts it. This is the opposite of an argument.

The root of his misunderstanding appears to be the mistaken belief that the "things mathematics are about" are in some way procedural in nature, and that when a mathematician writes down something that would seem to encode a nonterminating process, his only option for reasoning about that object is to somehow complete the nonterminating process. In fact, he can just…use the properties that define the object he has notated, without ever actually completing a supertask. To prove that every natural number is either odd or even, I need not check every natural number. Nevertheless, my deductions will hold for all (infinitely many) natural numbers. Mathematics is not computation, and computation is not mathematics. Specifications and algorithms are distinct.

I would expect anyone to actually take the time to learn something about the things he criticizes before going out and trying to solicit people's email addresses and (presumably) money. His ends are too noble for him to tarnish the entire concept of stateless research by abject ignorance about the subjects he purports to improve. He doesn't even have to believe in infinite sets; there are a lot of finitists who do great work and whom I respect. But those finitists necessarily know the first thing about logic, because they actually perform mathematics, and as such, don't attempt to make the claim that infinite sets are logically contradictory. They simply claim that they don't exist for non-logical reasons, and therefore infinite mathematics is valid deduction from false premises, i.e. it isn't "about" anything useful. If infinite sets were actually logically contradictory, especially via a 4-step deduction, software capable of automated proof generation would have found the contradictory sentences 50 years ago.

His general point stands, though. People writing introductory statistics textbooks for life and social scientists have completely bastardized the (mostly fine, in my opinion) foundations of statistics. The epistemological vices that flow from the "null ritual" (see Gerd Gigernzer's work) cause an eternal 95%-confidence lake of fire in which the non-physical sciences have been locked for a century. He is, however, too bold in extrapolating this observation. It's not a flaw native to higher levels of the Comte's hierarchy and gets passed down that causes this—it's a genuinely original mutation, adapted to these disciplines' students' mathematical and philosophical ineptitude, had I to guess. ² Hopefully, Patterson can do some self-reflection, improve his methodology, and produce a system that produces genuine knowledge and can actually out-compete the corrupt, statist hellhole we're all running from.