Bertrand Russell’s infinite sock drawer
physicstoday.scitation.org
physicstoday.scitation.org
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...
I think you mean 20th.
Why is intuitionistic reasoning blessed as the "universal reasoning," when there are other ways of doing logic that permit even fewer inference rules?
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.
[0] https://en.wikipedia.org/wiki/Self-reference#In_logic,_mathe...
[1] https://en.wikipedia.org/wiki/Russell%27s_paradox
[2] https://en.wikipedia.org/wiki/History_of_mathematics#19th_ce...
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.
Whereof one cannot speak, thereof one must remain silent.
Why can't we go from conclusions to premises?
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.
The difference is the order/sequence in which the events take place.
Regular maths starts with premises then looks for conclusions.
Reverse maths starts with conclusions then looks for premises.
So in reverse maths the premises follow from the conclusions - quite literally.
> Even if you go in reverse, finding premises for your conclusions, your conclusion must still follow from the premises you found.
GP means 'follow' in the sense of logical deduction, not follow in time.
Having found the premise (after the conclusion), the conclusion (we started with) must then logically follow from the premises (we later found).
The erudite/formal lingo aside. Reverse mathematics is a nice metaphor for how justification works in practice.
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.
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"?
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.
Symmetrical (equational) theories contain none.
Information mandates asymmetry.
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.
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.
[1] https://en.wikipedia.org/wiki/Axiom_schema_of_specification
[2] https://en.wikipedia.org/wiki/Axiom_schema_of_specification#...
Huh, I haven't thought about it that way. Very interesting.
Or the Philosophical cliche... I freely believe in the absence of free will.
Scott Aaronson has discussed this in more detail: https://www.scottaaronson.com/democritus/lec18.html
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”
There are the axioms of finite choice; and the axiom of infinite choice.
Broadly it's the philosophical distinction between Finitists and Infinitists.
Edit: or to put it more sharply: after one has seen the hilbert hotel the BT paradox is not surprising anymore.
It's mostly intuitive to computer scientists that infinities don't exist ;)
And we are also in the habit of proving/realizing our choice-functions rather then assuming them axiomatically.
https://en.wikipedia.org/wiki/Renormalization
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.
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.
∞
Basically, it doesn't hold for any constructive logic.
https://mathworld.wolfram.com/Banach-TarskiParadox.html
It is an existence proof that relies on the Axiom of Choice, of course there is no algorithm to actually construct those pieces.
This is one of the reasons that people distinguish between ZFC and ZF (Zermelo-Fraenkel with and without AC).
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.
B-T is Hilbert Hotel plus Axiom of Choice.
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).
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.
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.
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.
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.
My favorite explanation is this one: https://www.irregularwebcomic.net/2339.html
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.
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
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.
Become a Finitist (can I interest you in some pamphlets to tell you more about my religion?)