Thoughts from another infinity skeptic

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A recent thread (Does the square root of 2 exist?) dealt with mathematician Norman Wildberger’s aversion to irrational numbers, which he finds suspect because of their infinite decimal representations. It also dealt with his skepticism regarding infinity in general. Commenter petrushka posted a link to a relevant Medium article by Carlos E. Perez, who is similarly averse to infinities in mathematics. I thought it was worth an OP, so here’s a link:

Infinity as a Conceptual Shortcut in Mathematics

I’m skeptical of Perez’s skepticism, but I’ll save my thoughts for the comments.

117 thoughts on “Thoughts from another infinity skeptic”

  1. “Screw it. Life is too short.”

    I good response to lots of things. You just have to let some people be wrong! πŸ™‚

  2. I woke up this morning and found that “the concept horse is not a concept” had become comprehensible overnight. Just goes to show that when you’re struggling to understand something, it often helps to take a break and let your subconscious mull over it.

    So now I’m sucked back in, and it’s on to the next Fregean problem: how to show that “3 β‰  Julius Caesar” is a true statement. I’m not kidding. It’s called “the Julius Caesar problem”. πŸ™„

  3. Very informally, here’s how the Julius Caesar problem arises and how Frege solves it:

    1. Frege starts with something that was later christened “Hume’s Principle”, which is basically the notion that you can tell whether two sets A and B have the same number of elements without counting them by matching each element of A with exactly one element of B and observing whether there are any unmatched leftovers in either set. If there are no leftovers, then the sets have the same number of elements, or equivalently, they are “equinumerous”.

    For example, consider the sets

        \[ \begin{aligned} A &= \{\text{cat}, \text{dog}, \text{platypus}\} \\ B &= \{\text{Obama}, \text{Trump}, \text{Biden}\} \end{aligned} \]

    We attempt to match the elements one-to-one. Here’s one possible mapping:

        \[ \begin{aligned} \text{cat} &\leftrightarrow \text{Trump} \\ \text{platypus} &\leftrightarrow \text{Biden} \\ \text{dog} &\leftrightarrow \text{Obama} \end{aligned} \]

    We succeeded, and there are no leftover elements in either set, so we know that the sets are equinumerous without having counted. The number 3 played no role.

    2. Frege wants to employ that principle to define the # operator, which when applied to a property tells you how many objects have that property. Symbolically, #F gives you that number.

    3. Hume’s Principle tells us that #F = #G if the set of objects having property F can be matched one-to-one with the objects having property G, with no leftovers.

    4. We now have a way of deciding whether two numbers produced by the # operator are equal (ie identical), but we don’t really know what those “number” thingies are. Without that knowledge, we can’t assert that a number does or doesn’t equal Julius Caesar. For all we know, #F and #G could both be Julius Caesar, in which case “#F = #G”. That’s the “Julius Caesar problem”, and it’s undesirable, obviously. We need some way of distinguishing a number from Julius Caesar (or from any other non-numerical object, such as a tuba or a parking ticket).

    5. Frege sets Hume’s Principle aside and tries a different approach, defining something he calls “Basic Law V”. It effectively says that two sets are identical (identical, not just equinumerous) if each contains exactly the same elements as the other.

    6. He then defines numbers as explained earlier in the thread. For example, 2 is defined as the set of all sets that contain two objects.

    7. Now we know what numbers are, and that allows us to see, for example, that 2 is not equal to Julius Caesar because 2 is a set and Julius Caesar isn’t.

    8. That solves the Julius Caesar problem, but it creates another problem. Basic Law V leads to Russell’s paradox, which ruins everything. Frege wrote:

    Hardly anything more undesirable can happen to a scientist than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the second volume was already in print.

    Russell’s paradox obliterated (to use the Trump administration’s favorite word these days) Frege’s magnum opus. I’ll describe the paradox in a future comment.

  4. It’s worth mentioning that Cantor used Hume’s Principle in deciding whether two infinite sets were of the same size (the same “cardinality”). He didn’t use the name “Hume’s Principle”, because that was a later appellation, but he did employ the principle.

    This principle famously flummoxed (and then angered) JoeG, leading to much hilarity. The question for Joe was whether the sets A = {0, 1, 2, 3, …} and B = {1, 2, 3, 4, …} were of the same size. He was adamant that they couldn’t be, because the second set lacked an element that the first set contained: namely, 0. In reality, their sizes are the same because we can find a one-to-one mapping and apply Hume’s Principle:

        \begin{align*} A &\leftrightarrow B \\ 0 &\leftrightarrow 1 \\ 1 &\leftrightarrow 2 \\ 2 &\leftrightarrow 3 \\ 3 &\leftrightarrow 4 \\ &\vdots \end{align*}

    Every element in A is matched to a single element in B, and vice-versa. There are no elements left over. Therefore the sets are of the same size, despite the strong intuition that B must be smaller since it is missing one element that A contains: the number 0.

    The same reasoning can be applied to show that A = {0, 1, 2, 3, …} and B = {0, 2, 4, 6, …} are of the same size:

        \begin{align*} A &\leftrightarrow B \\ 0 &\leftrightarrow 0\\ 1 &\leftrightarrow 2 \\ 2 &\leftrightarrow 4 \\ 3 &\leftrightarrow 6 \\ &\vdots \end{align*}

    There’s a one-to-one mapping between A and B with no elements left over, so they are of the same size. This, despite the intuition that half of the elements are missing from B, so that it must be half the size of A. Infinite sets don’t work that way.

    Hume’s Principle can be used to show that all of the following sets are of the same size (or “cardinality”):

    the natural numbers
    the integers
    the even numbers
    the odd numbers
    the fractions (aka the rational numbers)
    the multiples of googol (1 \times 10^{100}, 2 \times 10^{100}, 3 \times 10^{100}, \cdots)
    the numbers n^{n^{n^{n}}} for n = 1, 2, 3, …

    Each of those sets can be placed into one-to-one correspondences with all of the others, so all are of the same size of infinity, which is referred to as \aleph_0.

    There are bigger infinities than this. Infinitely many, in fact.

  5. On to Russell’s paradox.

    First a note: sets technically don’t exist in Frege’s system. Instead there are these things called “extensions” which act a lot like sets but are subtly different. For the purposes of these comments about Frege’s system, I’ve been glossing over the distinction because it hasn’t mattered for my explanations. It won’t matter for my description of Russell’s paradox, either, but I thought I should mention it.

    Sets can have other sets as elements (and in Zermelo-Fraenkel set theory, which is generally considered the foundation of all mathematics, every object is a set. Even numbers are sets). Sets can even have themselves as elements, which leads to weird infinite regresses when you try to write them out (which is probably one of the reasons the infiniphobic Normal Wildberger dislikes standard set theory). Despite the infinite regresses, they are logically consistent and don’t break anything.

    The set of all sets that contain themselves is an unusual but legitimate set. The set of all sets that don’t contain themselves isn’t legitimate, though, because it creates a paradox.

    To see why, ask yourself: does the set of all sets that don’t contain themselves contain itself? If it doesn’t contain itself, that means it belongs to the set of all sets that don’t contain themselves, which means that it contains itself. In short, if it doesn’t contain itself, it contains itself.

    Flipping it around, if it does contain itself, that means that it doesn’t belong to the set of all sets that don’t contain themselves, which means that it doesn’t contain itself. In short, if it contains itself, it doesn’t contain itself.

    It’s a contradiction either way, and if your system leads to a contradiction, it’s inconsistent and therefore broken. Modern set theory eliminates the problem by outlawing sets of the kind that lead to the paradox, but Frege’s system was fatally wounded by it.

    There’s a real-life version of Russell’s paradox that makes it easier to picture. Imagine a barber in a small town who shaves every man who doesn’t shave himself. Does he shave himself? If he does, then he isn’t a man who doesn’t shave himself, which means that he doesn’t shave himself. In short, if he shaves himself, he doesn’t shave himself.

    Flipping it around, if he doesn’t shave himself, then he is one of the men who don’t shave themselves, which means that he shaves himself. In short, if he doesn’t shave himself, he shaves himself.

    Same principle as with the sets, but easier to picture.

  6. It’s worth explaining how the natural numbers are “constructed” in standard set theory these days, because ZFC set theory (Zermelo-Fraenkel with the Axiom of Choice) is widely regarded as the foundation of mathematics (though it isn’t the only possible foundation).

    It’s a brilliant yet simple scheme. Each number is defined as the set of all lesser numbers. So 7 := {0, 1, 2, 3, 4, 5, 6}, and 3 := {0, 1, 2}. 0 is defined as the empty set, symbolized as {} or βˆ….

    Working up from zero, that means that

        \[ \begin{aligned} 0 &:= \emptyset \\ 1 &:= \{0\} = \{\emptyset\} \\ 2 &:= \{0, 1\} = \{\emptyset, \{\emptyset\}\} \\ 3 &:= \{0, 1, 2\} = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\} \\ 4 &:= \{0, 1, 2, 3\} = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}, \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}\} \end{aligned} \]

    And since βˆ… is just the empty set, {}, the above can be rewritten as

        \[ \begin{aligned} 0 &:= \{\} \\ 1 &:= \{ \{\} \} \\ 2 &:= \{ \{\}, \{ \{\} \} \} \\ 3 &:= \{ \{\}, \{ \{\} \}, \{ \{\}, \{ \{\} \} \} \} \\ 4 &:= \{ \{\}, \{ \{\} \}, \{ \{\}, \{ \{\} \} \}, \{ \{\}, \{ \{\} \}, \{ \{\}, \{ \{\} \} \} \} \} \end{aligned} \]

    It’s sets all the way down, ending in… nothing. No primitives other than sets are needed. That’s brilliant. The successor function, which is the equivalent of adding 1 to a number, is simply

        \[ S(n) := n \cup \{ n \} \]

    …where ‘βˆͺ’ is the union operator. Who was it who came up with this scheme, you ask? None other than the legendary John von Neumann, and he did it at age 19, lol.

    He isn’t a household name, but I think he deserves to be. He’s arguably in the same class as Newton and Einstein, and considerably above Hawking, who is a household name.

    Von Neumann made key contributions to math, physics, computer science, economics, and the Manhattan Project. He could converse in ancient Greek by age 8 and was fluent in Hungarian, German, French, and English. He invented the von Neumann architecture, which was a milestone in my field of computer engineering. He was a lightning-fast mental calculator and had a photographic memory. There are stories of him memorizing pages of the phone book and reciting them backwards. He was also very funny and an avid partyer. His friends called him Good Time Johnny.

    He died far too soon of cancer at age 53, likely because of radiation exposure during the Manhattan Project. You have to wonder what he would have accomplished had he lived 20 more years.

  7. I looked up the axioms of ZFC set theory. Note that these are just the axioms — it takes tons of definitions to construct math on top of ZFC (including the definitions of the natural numbers that I explained in the previous comment). The key point is that everything beyond the axioms β€” including the construction of numbers β€” consists of definitions and derivations, not additional assumptions. The axioms are the only foundational assumptions needed.

    A mere nine axioms at the foundation of all of mathematics. That’s wild.

    Here are the axioms in symbolic form. I’ll provide translations, first literally, and then in an easier-to-understand form.

    1. βˆ€A βˆ€B [βˆ€x (x ∈ A ↔ x ∈ B) β†’ A = B]

    For all sets A and B, if for all x where x is in A it’s also in B, and vice-versa, then A = B.

    A = B if all of their elements are the same.

    2. βˆƒA βˆ€x (x βˆ‰ A)

    There exists a set A such that for all x, x is not in A.

    There is an empty set.

    3. βˆ€A βˆ€B βˆƒC βˆ€x [x ∈ C ↔ (x = A ∨ x = B)]

    For any two sets A and B, there exists a set C such that for all x, if x is in C, then x = A or x = B.

    For any two sets A and B, there is a set C containing both of them and nothing else.

    4. βˆ€A βˆƒB βˆ€x [x ∈ B ↔ βˆƒC (x ∈ C ∧ C ∈ A)]

    For every set A, there exists a set B such that for all x, if x is an element of B it means that there exists a set C where x is in C and C is in A, and vice-versa (whew!).

    The union of all the sets contained within A exists. That is, we can construct a set B which contains all of the elements of all of the sets contained within A.

    (You might wonder, as I did, why the axiom doesn’t deal with a simpler two-way union which we could generalize to an n-way union. I think it’s because you’d need an extra axiom or two to support the generalization, but we don’t want that because we’re trying to minimize the number of axioms.)

    5. βˆ€A βˆƒB βˆ€x [x ∈ B ↔ x βŠ† A]

    For every set A, there exists a set B such that for every x, if x is in B, x is a subset of A, and if x is a subset of A, it is also in B.

    Every set A has a corresponding power set B, which is a set containing all possible subsets of A.

    6. βˆƒA [βˆ… ∈ A ∧ βˆ€x (x ∈ A β†’ x βˆͺ {x} ∈ A)]

    There exists a set A such that the empty set is in A and for every set x in A, the union of x with {x} is also in A.

    You can construct an infinite set by repeatedly taking the union of a finite set with the set containing it — x βˆͺ {x}.

    This one actually disappointed me, not because there’s anything wrong with the axiom but because it predated von Neumann and it invoked the pattern upon which he ended up basing his definition of the natural numbers. Don’t get me wrong — he was still a genius, and his realization that he could use that pattern as the basis for the natural numbers was brilliant. It’s just slightly less impressive that the 19-year-old von Neumann didn’t invent the pattern ex nihilo, but instead observed it and exploited it for his own purposes.

    7. βˆ€x ∈ A βˆƒ!y Ο†(x, y) β†’ βˆƒB βˆ€y [y ∈ B ↔ βˆƒx ∈ A Ο†(x, y)]

    For every element of a set A, there exists exactly one y such that if a function Ο† maps x to y, it means there is a set B such that for any y, if y is in B, it implies that there is an x in A such that the function maps x to y, and vice-versa.

    (A note regarding the exclamation point: βˆƒ!y means “there exists exactly one y”, while βˆƒy means “there exists at least one y”.)

    You can apply a function to all the elements of a set and form a set out of the results. In technical terms, the image of a set A under a function Ο† is also a set.

    8. βˆ€A [A β‰  βˆ… β†’ βˆƒx ∈ A βˆ€y (y ∈ x β†’ y βˆ‰ A)]

    For every set A, if A isn’t the empty set, it means that there exists an x which is an element of A such that for all y, if y is a member of x then y isn’t a member of A.

    This is a clever way of disallowing sets that are subsets of themselves, and it avoids paradoxes such as Russell’s.

    9. βˆ€A [βˆ€x ∈ A (x β‰  βˆ…) β†’ βˆƒf: A β†’ βˆͺA βˆ€x ∈ A (f(x) ∈ x)]

    For every set A such that no element of A is the empty set, there exists a function that maps every element of A to an element in the union of all the sets of A, subject to the proviso that for each element of A, the function maps that element to an element of that element. Lol.

    For any set A of non-empty sets, there exists a function f capable of choosing one element from each subset of A.

    Even more informally, if there are a bunch of open boxes on the table in front of you, each containing at least one stuffed animal, then there is a way of choosing one stuffed animal from each of those boxes. Sounds obvious, but here’s the catch: you can’t just reach into a box and grab an animal at random. Randomness doesn’t exist in the system. There has to be a rule that you’re following that determines which animal you choose from any given box.

    If you’re dealing with a finite number of boxes, there is always a rule that will meet this requirement, and that’s true even without the axiom. However, if the number of boxes is infinite, there is no guarantee that such a rule exists. So the axiom steps in and declares that such a rule exists.

    This axiom is known as the Axiom of Choice. It’s a simple axiom, but it has far-reaching and sometimes bizarre consequences that I might explain in a future comment.

  8. Reading about von Neumann’s achievements, I learned that he invented “von Neumann algebras”, which are operator algebras defined over Hilbert spaces. Hilbert employed these spaces in his work, but it was von Neumann who coined the term “Hilbert space”.

    There’s a funny story (most likely apocryphal) about a von Neumann lecture in which he mentioned Hilbert spaces. Hilbert was in attendance, and at the end of the lecture he raised his hand and asked “What’s a Hilbert space?”

    Hilbert spaces are vector spaces which can be infinite-dimensional. The irony is that Hilbert, a finitist, employed these infinite-dimensional objects, and not only that — they also ended up being named after him. That’s on top of the fact that he admired Cantor’s work on infinities and fondly spoke of “Cantor’s paradise”. For a finitist, he sure seemed to have an affinity for infinities.

    I suspect he was a closet infinitist but was too ashamed to confess to that vice. Describing infinities as useful fictions allowed him to have his cake and eat it too.

  9. “Describing infinities as useful fictions allowed him to have his cake and eat it too.”

    That is, it’s a useful fiction to be able to dismiss the nature of infinity as a useful fiction

  10. Having studied the ZFC axioms, I thought it would be interesting and possibly entertaining to see what Norman Wildberger thinks of modern set theory, with its infinite number of infinities.

    He is not a fan. The title of one of his videos is “Modern “Set Theory” – is it a religious belief system?” He really seems to believe that set theory is a matter of faith.

    In that video, he claims:

    1. Set theory is logically inadequate as a foundation of mathematics and should be supplanted by natural number arithmetic. Set theory should be regarded not as foundational but as a branch of combinatorics.

    2. As we’ve seen in previous threads, he thinks math should be tied intimately to computer science:

    We want to be in sync with what our machines, what our computers are doing.

    Two problems with that: 1) there’s no reason to restrict mathematics to what computers can do — that’s an artificial limitation; and 2) computers aren’t limited in the way he seems to think they are. With the right software, computers can do math symbolically and can (for example) represent \sqrt{2} as “\sqrt{2}” instead of as an approximation.

    3. He claims that the continuum is problematic because irrational numbers don’t exist. I’ve already dealt with that in other threads, so no need to reinvent the wheel here.

    4. He invokes the same method as aleta for locating \sqrt{2} on the x-axis: draw a right isosceles triangle with sides of 1 at the origin, and then rotate the hypotenuse so that it lines up with the x-axis. One end is at the origin and the other is at \sqrt{2}.

    Except that for Norman, \sqrt{2} doesn’t exist, so the hypotenuse doesn’t have a length. It’s bizarre, but he truly believes it, referring to the hypotenuse as a “segment” that “doesn’t necessarily have a length” and adding that “we want to believe in such lengths”, as if it were an article of faith.

    In Wildberger World, some line segments have lengths and others don’t.

    What’s especially goofy is that as faded_Glory pointed out, you can take that same hypotenuse and label the endpoint with a ‘1’ instead of ‘\sqrt{2}‘, thus creating a different set of units. Now suddenly it’s the hypotenuse that has a length while the sides don’t. The hypotenuse does and doesn’t have a length, depending, and the sides do and don’t have lengths.

    Wildberger just needs to get over himself and accept that all line segments have lengths, even if those lengths can’t be

    5. He speaks of the “conundrum” of setting up an arithmetic that includes both rational and irrational numbers, saying that it’s “hugely problematic”, and he blames this nonexistent conundrum for “almost all of the difficulties with modern calculus, and modern analysis, and ultimately with modern set theory too.”

    In other words, this misconceived conundrum leads Wildberger down his garden path of errors.

    Why does he think that arithmetic becomes problematic when the irrational numbers are included? He doesn’t say in this video, but from what he’s said elsewhere, his objection appears to be that we can’t write out the results of arithmetic operations in decimal form when they include the irrationals, just as we can’t fully write the irrationals themselves in decimal form. He’s once again mistaking a limitation in representability for a limitation in arithmetic itself.

    To be continued…

  11. Correction to the above:

    Wildberger just needs to get over himself and accept that all line segments have lengths, even if those lengths can’t be expressed in a given system of units using rational numbers.

  12. keiths: The title of one of his videos is β€œModern β€œSet Theory” – is it a religious belief system?”

    I watched that video. He complains that mathematics becomes a branch of philosophy if it relies on ZFC. Then I checked Wildberger’s youtube channel, and it seems to be mostly philosophy. Strange.

    I can agree with some of his criticisms of set theory. But so what?

    For myself, I’m a pragmatist. Classical mathematics works well. Much of it has arisen from the needs of physics. His own version of intuitionistic mathematics does not accommodate physics nearly as well.

  14. Neil:

    I watched that video. He complains that mathematics becomes a branch of philosophy if it relies on ZFC. Then I checked Wildberger’s youtube channel, and it seems to be mostly philosophy. Strange.

    It’s ironic, isn’t it? He even complains about mathematicians “outsourcing” the foundations to the philosophers, as if that were a bad thing. Is it a dereliction of duty? Does the mathematician’s union (Local 51, AFL-CIO) prohibit mathematicians’ jobs from being taken by philosophers?

    Plus, it’s not like mathematical logic is some airy-fairy branch of philosophy. I mean,

        \[ \exists F \exists a \left( F a \land x = \# u (F u \land u \ne a) \land y = \# u F u \right) \]

    is pretty damn mathematical. Axioms, definitions, derivations, theorems. All very mathy.

  15. Continuing this comment.

    6. Of ZFC, he says:

    What we ended up having was a framework which was not really based on precise definitions and clear theorems and such, but was an axiomatic framework.

    Why he thinks that axiomatic systems can’t be based on precise definitions and clear theorems is a mystery to me. Axiomatic systems are all about precision and clarity! I’d like to see him specify what’s imprecise and unclear about ZFC. ‘Axiomatic’ seems to be a dirty word for him in the same way that ‘infinite’ is. It’s pure prejudice.

    7. He evidently feels that his own conception of mathematics as founded on the natural numbers is somehow a non-axiomatic system, but why? He treats the natural numbers and their operations as givens. They’re primitives, not derived from anything else. How is that not axiomatic? He may not identify his axioms as such, and he may not even be aware that they’re there, but they are most definitely axioms.

    8. He writes:

    It was as if we had abandoned the effort to try to set up this foundational issue very carefully and clearly and said “Let’s just assume that it works”.

    That statement is bizarre. How does the careful development of ZFC as a foundation amount to an “abandonment” of the effort to establish a solid foundation? And “let’s just assume that it works”? Doesn’t he realize that ZFC has been thoroughly poked, prodded, and tested to make sure that it works? Doesn’t he realize that if ZFC didn’t actually work, mathematicians would have ditched it and developed something better, in the same way that they abandoned Cantorian set theory when its problems became apparent and embraced ZFC instead?

    9. He claims that there is “a polite agreement not to examine carefully what this logical foundation of set theory is.” There is no such agreement, and anyone is free to examine that foundation.

    10. He likens the mathematicians’ supposed decision to “outsource” the foundations of math to the philosophers to IBM’s decision to outsource the development of the PC’s operating system to Microsoft. Both were mistakes, according to Wildberger, because in both cases what was being outsourced was outsourced only because it was considered to be unimportant. That’s ridiculous. Mathematicians didn’t say “Nobody cares about foundations. Let’s let the philosophers deal with it”. And as I remarked earlier, this weird tribal distinction he’s drawing between mathematicians and mathematical logicians is arbitrary and not really defensible.

    11. Wildberger claims that students are told

    Don’t go there. There are these foundations. You just believe what you’re told. Play along with the game, and we’re going to concentrate on doing sophisticated important mathematics. Foundations are unimportant.

    He adds:

    Well, my view is exactly the opposite… I think that more and more complicated mathematics is much less important than getting the foundations correct.

    That’s a strawman. The view among mainstream mathematicians isn’t that the foundations are unimportant; it’s that the foundations have already been established, so that we can move on to other problems. Mathematicians don’t see the imaginary flaws that Wildberger thinks he sees, and they aren’t trying to fix what isn’t broken.

  16. keiths: Why he thinks that axiomatic systems can’t be based on precise definitions and clear theorems is a mystery to me.

    I’m not into reading minds. But it looks as if Wildberger is particularly concerned about ontology — about what mathematical objects are.

    Personally, I don’t care about ontology, which is why I can be a fictionalist. What matters to me, is what operations can be done on mathematical objects. And axioms are a good way of defining those.

  17. Neil:

    But it looks as if Wildberger is particularly concerned about ontology β€” about what mathematical objects are.

    Well, he insists that they are finite constructions, and in that way his ontology places unnecessary limits on his mathematics.

    Personally, I don’t care about ontology, which is why I can be a fictionalist.

    But to be a fictionalist is to take an ontological stance. You believe (as I do) that mathematical objects are useful fictions, created by us. They don’t exist in some separate platonic realm beyond time and space, and that is an ontological claim.

    What matters to me, is what operations can be done on mathematical objects. And axioms are a good way of defining those.

    Yes, and wittingly or unwittingly, Wildberger relies on axiomatic claims about the natural numbers even if he doesn’t state them formally as axioms.

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