The article provides a nice introduction to the subtleties surrounding the axiom of choice. The last paragraph briefly mentions that there are other flavors of mathematics in which the axiom of choice doesn't hold. For people who are intrigued by the idea of different flavors of mathematics, but all still set-based, a couple months ago I wrote an expository summary of the so-called "multiverse philosophy in set theory". You can find it at https://iblech.gitlab.io/bb/multiverse.html, and I'm happy to answer any questions or comments you might have.
There are also even wilder flavors of mathematics which are not set-based. In those we can have various dream results which directly contradict the mathematical canon (but are internally consistent and have a certain precise relation to the ordinary mathematical world). For instance we can have that every function is computable by a Turing machine, that every real function is continuous or that the reals include infinitesimal numbers. An introduction to these flavors, aimed at philosophers of mathematics, can be found here: https://rawgit.com/iblech/internal-methods/master/paper-film...
Good question! I would disagree that intuitionistic reasoning is blessed as the "universal reasoning". As you say, there are other even weaker but still useful systems around.
It is just a fact of life, without any room for philosophical preferences, that the largest common denominator of all toposes is exactly intuitionistic reasoning, not more, not less.
But there are, besides toposes, also other kinds of mathematical structures which can be regarded as mathematical universes! And the largest common denominator of those other kinds can be more or less than intuitionistic reasoning.
Going up, we have for instance models of ZFC. By definition, their largest common denominator is ZFC, so more than intuitionistic reasoning.
Going down, we have the so called "arithmetic universes". Their largest common denominator is "arithmetic type theory", a predicative flavor of intuitionistic reasoning. (To a very rough first approximation which doesn't at all do justice to this intriguing topic, "predicative" means "no powerset axiom". The terminological convention is that by default, "intuitionistic reasoning" refers to impredicative intuitionistic reasoning.)
And then there are a couple other kinds still.
That said, toposes with their impredicative intuitionistic reasoning do occupy a sweet spot. They are sufficiently general to yield useful applications in several branches of mathematics while not being too general.
Self-reference in logic is named after the ancient symbol Ouroboros, a dragon that continually consumes itself, denotes self-reference [0]. I found out about this when reading about Russell's paradox [1]. The turn of the 19th century was an extraordinary time where logicians, mathematicians and philosophers "discovered" revolutionary ideas that we are still today trying to understand and find proofs [2].
The circularity supported in programming, that programs ("quines") can refer to their own source code, illustrates the same issue at heart. Yusuke Endoh created a tantalizing Ouroboros quine cycling through 128 languages: https://github.com/mame/quine-relay
I'm always happy to see solid philosophy show up here. Really, it's an under-appreciated field that gets written off as useless. Looking back at when I was in college, one of the courses that I still make use of was a philosophical logic class. Infact, the prof I had for that course sent me a link to the Open Logic Project, which has come in handy ever since.
I also have a soft-spot for Russell and his student Wittgenstein. Tractatus is an incredible, though later redacted, work of pure axiomatic reasoning. While HN focuses mostly on tech, I think that the kind of reasoning found in Analytic philosophers can be a boon to anyone doing anything that requires the sort of logical design found in the technology field.
The Tractatus is couched in language that make it seem like Wittgenstein is laying out a mathematical proof, but many of his conclusions don't follow from his premises. The Tractatus is much more a work of mysticism (in the religious sense) than of logic.
Whereof one cannot speak, thereof one must remain silent.
Even if you go in reverse, finding premises for your conclusions, your conclusion must still follow from the premises you found.
Saying that the premises don't follow from the conclusions means that, taking the premises as true, the conclusion is may or may not be true, so it is illogical to draw that conclusion from those premises. Or if you prefer the other way around, if, taking the conclusion as true, the premises could be true or false (or taking the conclusion as false, the premises could still be true or false) then the conclusion does not follow from the premises you found.
I am, really, and the idea of reverse maths/logic seems very interesting.
I was just pointing out that the GP's use of the word 'follow' was not about the temporal order of how discoveries are made, but to the logical concept of implication.
That is to say, the GP wasn't complaining that the Tractatus is doing reverse mathematics. They were complaining that the Tractatus is presenting illogical arguments, that it is taking logically unrelated statements and presenting them as conclusions and premises.
I am not sure how to even respond to this coherently... alas - I try.
Do you think "logical implication" (whatever that is) is not bound by temporal order?
That simply tells me that whatever you think "logic" is - it doesn't concern itself with time or downward causation. e.g your idea of "logic" is not Linear/Temporal logic.
So it can't be the logic of this universe then? Perhaps you've heard the saying "One man's modus ponens is another man's modus tollens"?
Yes, a is implied by b is equivalent to me to b implies a, and it is an atemporal relationship. This is how a lot of logic is taught and practiced, whether in mathematics, physics, engineering or programming. I am a programmer by trade, and even in programming most uses of logic have no concept of time.
When I say x + 1 = 7, therefore x = 6, I see the two statements as being true simultaneously, and simultaneously with the implication.
I am sure there exist logics where time is a necessary component of reasoning, and I am not downplaying their importance. But there also exist logics where time plays no part, and they are not more or less true.
> Even today it’s regarded with suspicion in a way that most mathematical axioms aren’t.
This is a common misconception. You either use it as an assumption, or you do not, as is the case with the parallel postulate. There need be no controversial sentiments. In the same way complex numbers were briefly "controversial", but as mathematicians we shouldn't bring too much opinion into the matter; we should only follow the argument. In the last paragraph the article seems to admits that the approach is you either assume it or you do not.
It's controversial in the way addition is not. It's controversial as to which axioms should be the common default in the language, and where research funding should be spent.
The way ZFC navigates around Russel's paradox is by adopting the axiom of restricted comprehension [1].
Here's a thought experiment: What happens if we allow for unrestricted comprehension [2] ? What happens if we say 'Contradictions exist and they are empirical. What do they mean?'
The upside is that you attain "unrestricted comprehension" (In the English, not Mathematical sense) with the miniscule downside of having to navigate around contradictions from time to time.
Contradictions exist - if they didn't I wouldn't be able to contradict myself when I want to. I wouldn't be able to trigger exception-handlers in your brain when I want to.
How you handle that exception is a matter of choice.
I like the Dialetheist solution [3]. Basically the Axiom of Unrestricted comprehension is akin to practicing the Principle of Charity.
Is that a contradiction or a lie? Genuine question: is there a difference between the two?
Edit: I read the article (and I’m not sure I was able to follow it completely) but it seemed to mostly be about free will, and not people making contradictions (or lying).
You are correct in that I am appealing (exploiting?) the Liar's paradox [1]. The gist of which is that the truth-value of the proposition is undecidable.
You could interpret my statement as a performative contradiction; or you could interpret it as a lie, but a far more interesting a conversation would ensue if you simply ask me "Why do you say that?"
Which is why I said that it's up to you on how you choose to handle the exception (which I have intentionally triggered in your brain).
The way I would prefer you to interpret my intentional contradiction is to see it for what it is. I am engaging in cooperative multi-tasking [2]. I am yielding control by triggering an exception. Your turn to steer the conversation.
“Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox”
I do not really understand why the Banach-Tarski paradox is such a big deal. It does not seem any different from the paradoxes of the Hilbert hotel. If one has two filled hilbert hotels one can fit all hotel guests in one single hilbert hotel by letting guests from one hotel take even numbered rooms and from the other hotel odd numbered rooms.
The Hilbert's-Hotel-style result that the union of two balls consists of exactly the same amount of points than just a single ball is much more basic than the Banach–Tarski paradox.
In the Banach–Tarski paradox, we cut a single ball into a finite number of pieces (incidentally, it can be done with just five). These pieces are then rotated and moved in space, but otherwise kept exactly as they are. The surprising fact is that after moving and rotation, the five pieces fit together to form two balls.
I don't understand Banach-Tarski, but it seems like the common version which you just explained must be wrong. It is not possible to double the volume of an object by cutting and reassembling. So someone has either made a mistake somewhere, or the "paradox" is using highly perverse definitions of "ball", "piece" etc such that if you understand what bizzare and unphysical concept was represented by them, the surprise goes away. The counterintuitive nature of the popular version is simply a lie, one way or the other.
You are spot on! The paradox uses a, from the point of view of everyday life, highly perverse definition of "piece".
The definition of "piece" used is: "Arbitrary collection of points which need to bear any relation." This is in contrast to pieces of things in everyday life, which are firstly not made of infinitesimal points, but even if we keep rolling with that, satisfy the restriction that with every included point also some neighboring points are included.
This is not so with the "pieces" appearing in the Banach–Tarski paradox. In fact, they are so extremely rigged/fractal/dislocated/non-contiguous, that it is not possible to assign a meaningful measure of volume to these. For instance, the unit cube has volume 1. The empty set of points has volume 0, and so do sets which contain just a finite number of infinitesimal points. But the "pieces" appearing in the paradox are so weird that we cannot meaningfully attach any volume at all to them.
Also, there can never be any formula or any other explicit description of which points the five pieces consist of. This is because the proof employs the axiom of choice.
> I don't understand Banach-Tarski, but it seems like the common version which you just explaing must be wrong. It is not possible to double the volume of an object by cutting and reassembling.
Well, it seems that you DO understand why it is considered to be very surprising and unintuitive.
The definitions of "ball" and "rotate" are perfectly normal. The definition of "piece" is slightly odd in that the "pieces" don't have smooth surfaces -- they are "jagged" or "fuzzy" down to an infinite level. (This is why common notions like "volume is conserved" don't apply -- such notions don't apply normally even to simpler objects like a fractal Serpinski Sponge.)
What is especially weird about the pieces (axiom-of-choice weird) is that specifying exactly what the boundaries of the pieces are is so hard that no clear description or algorithm can be given that specifies them. It's sort of like if you said "cut a ball so the prime-numbered points are in piece 1 and the composite points are in piece 2", except MORE strange because the definition of "prime" is easy to understand.
> This is why common notions like "volume is conserved" don't apply -- such notions don't apply normally even to simpler objects like a fractal Serpinski Sponge.
No, volume always is conserved just fine, as long as you deal with measurable pieces. Sierpinski’s sponge has a well defined volume, and this volume behaves in a sensible manner. Normal sets have well defined volume.
The thing with Banach-Tarski pieces is that they cannot be assigned any volume in any sensible way. It has nothing to do with them being “jagged” or “fuzzy”, but rather with their weird behavior when it comes to self-overlapping translations.
The core of the matter is in the word "cutting". It is possible to pick points off of a connected sphere in such a way that you have 5 different collections of points on the surface of the original sphere (they are NOT connected surfaces). You then take these 5 sets, apply rotations and translations, and at the end, you have 2 spheres of identical volume. During the process, no 2 points from the same piece have changed their relative positions (of course, points from different pieces have).
The 5 pieces are highly un-intuitive objects. I imagine if you were to realize even an approximation of them, they would each look like a complete sphere, with infinitely many infinitesimally small holes poked through. But when you do put them together, each and every one of the (uncountably) infinitely many holes in the 1 sphere is perfectly plugged by one of the (uncountably) infinitely many points in one of the other spheres.
One point regarding whether or not this is unphysical is that in physical systems, conceptually we work in finite systems and at the only very end of our calculations we take the infinite limit while holding physically observable quantities fixed. This is the essence of the thermodynamic limit.
In these paradoxes, infinities are present from the outset and I think it's this that leads to the unphysical outcome. They're not wrong. They are mathematical paradoxes. But it's not a problem they are unphysical (from a physics point of view) because physics uses mechanisms, like the thermodynamic limit, to handle infinite limits sensitively. Then the paradox goes away.
For instance, the 'physics' version of the Hilbert hotel problem would say there are two hotels, one with N rooms and the other with M rooms. Then do all the renumbering you like, the paradoxical situation of filling both hotels and then putting all guests from both hotels into one of them is no longer possible. Finally, if you want to think about hotels with an infinite number of rooms take N and M to inifinity keeping N/M fixed
Edit: add physicified version of Hilbert hotel problem
They're quite different. The Banach-Tarski paradox requires a finite number of pieces and physically simple transformations. It's not simply the discovery of a bijection between a sphere and two spheres.
Yes, but I think the basic cause of both is that one assumes there are infinite sets. The Banach-Tarski paradox to me seem to be little more than a three dimensional variation of the hilbert hotel. Therefore, if one wants to implicate any basic axiom for being the cause of the counter-intuitiveness of that theorem the infinity axiom seems the more sensible candidate. Therefore, to me, it seems the argument against the axiom of choice from the Banach-Tarski paradox does not make any sense.
I don't really think so. I am arguing that whatever axiom of choice one uses is basically an implementation detail of the banach-tarski paradox and that because of the similarity to the hilbert hotel paradox the cause of the paradox should be sought in the infinity axiom and not in whatever axiom of choice one prefers or doesn't prefer.
Edit: or to put it more sharply: after one has seen the hilbert hotel the BT paradox is not surprising anymore.
Well, if one assumes finiteness all of these paradoxes go away and this discussion is not about anything anymore. I studied physics and am quite attached to infinities. That one constrains oneself to a finite state system mostly does not mean that one is not allowed to look out of the window towards the stars and marvel at infinity anymore.
If giving up infinities allows me to understand The Universe better, I don't see how that robs me of marveling at the stars - it only makes it more exciting!
I can't comprehend an infinite universe. Nobody with finite memory/time can.
Well, I did a PhD in quantum field theory and as part of that published a paper on renormalization..... Of course the infinite sets involved can be approximated using finite sets if one does not mind losing important symmetries. Still, these finite sets would still be frightingly and incomprehensibly large.... If one wants to comprehend anything one may be better off just ignoring quantum mechanics..... And I am not talking about the measurement problem. I am talking about the incredible vastness of the state space. The wave function of the universe is an incredible and completely untractable thing even if one assumes it is finite. It contains all possible histories of the universe for one thing and lets all of these possible histories interact with one another.
I hear you (and I obviously made a bad inference re: your background - apologies), but there is a distinction to be made between large-but-finite (computable? halting?) and infinite (non-computable? non-halting?) sets.
We can't comprehend the universe using numerical methods - it's too complex for our brains/computers.
But we can understand complexity using symbolic methods.
Symbolism/representationalism (religion) is rather inevitable part of the human condition.
Infinite sets are a useful abstraction, but uncountable infinities can probably be avoided since the Löwenheim-Skolem[0] theorem says that any countable first-order theory has a countable model.
> ZFC is not regarded as sound by some because of these Paradoxes.
Can you give relevant pointers? I'm curious because while I do know a couple of arguments for the inconsistency of ZFC, none of these are related to the Banach–Tarski paradox or similar nonintuitive results.
Also I'd like to stress that ZFC and ZF are exactly as consistent: If ZF is consistent, then so is ZFC (and vice versa). This meta-result is proven in a very weak finitary logical meta-system ("PRA"), hence can be trusted even if one is wary of set-theoretic infinities. The keyword is "Gödel's constructible hierarchy".
In this line, there are even more astonishing meta-theorems, all provable in PRA:
* From any ZFC-proof of a purely number-theoretical statement (a statement which refers only to natural numbers), any usage of the axiom of choice can be mechanically eliminated. That is, any ZFC-proof of such a result can be transformed into a ZF-proof.
* From any ZFC-proof of a statement of the form "for all numbers n, there exists a number m such that ...", where in "..." no more quantifiers ("for all", "there exist") appear, any usage of the law of excluded middle ("any statement is either true or not") can be mechanically eliminated. That is, any such ZFC-proof can be transformed into a proof in IZF ("Intuitionistic Zermelo–Fraenkel"). The keyword is "double negation translation".
These meta-theorems indicate that the axiom of choice and the law of excluded middle can be regarded as "mathematical phantoms". Just as the complex numbers work wonders for us but can be compiled away to pairs of real numbers, these set-theoretic principles promise to work wonders for us and their usage can be compiled away if we wish so.
The construction involves a rotation of an infnitely detailed object in a way which is analogous to Hilbert Hotel. (Rotating it makes it fill empty space)
Perhaps it is unsurprising, but it remains relevant. In a set theory that allows infinite sets, but does not allow the axiom of choice, you can construct the Hilbert hotel, but not the Banach-Tarski paradox, if I understand correctly.
On the other hand, to me it sounds like the distinction may also come from the difference between countable infinities (which Hilbert's Grand Hotel is limited to), and uncountable infinities (which BT depends on).
Logicomix is the best. I strongly second the recommendation.
This novel not only discusses the logical aspects of self-referentiality, it also goes strongly meta on it. At one point in the novel, you see the authors of the novel debating on how to best present a particular story.
There are also even wilder flavors of mathematics which are not set-based. In those we can have various dream results which directly contradict the mathematical canon (but are internally consistent and have a certain precise relation to the ordinary mathematical world). For instance we can have that every function is computable by a Turing machine, that every real function is continuous or that the reals include infinitesimal numbers. An introduction to these flavors, aimed at philosophers of mathematics, can be found here: https://rawgit.com/iblech/internal-methods/master/paper-film...