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benrbray
voldacar
Such breadth, you could dedicate a lifetime to learning only one of those topics!
amelius
I tried to access "Differential Geometry and Lie Groups" but it seems there is only the table of contents.
throwaway81523
Wow! There is a lot there to read.
throwaway81523
I'm looking at this now. It's a bit idiosyncratic but looks good. It covers standard topics in proof and recursion theory but from a more contemporary approach using natural deduction instead of Hilbert-style proof systems, and it also does complexity theory with various computation models. There are a fair number of exercises and problems, always a good thing. But it covers specialized topics at the expense of some standard ones. For example, I didn't spot any mention of the compactness theorem. So it's hard to call the book a usable general introduction to logic, despite it starting at a relatively beginner level.
It's a draft and it has some typos and some long-winded passages that could use editing, but I'm glad it is out there, and it looks to me like a good way for computer people to learn some logic and computability theory if that strikes their fancy. I've only looked at the first couple chapters but the TOC mentions that Girard's polymorphic lambda calculus makes an appearance, so there must be some type theory in there. But, my preference would be to increase the coverage of that even if it means trimming back some of the computability stuff. Another improvement might be to include some coverage and exercises in mechanized proof checking using Coq, in the spirit of Software Foundations (also by authors at Penn, https://softwarefoundations.cis.upenn.edu/ ).
Thanks to whoever posted this link!
harperlee
Could you provide a couple of alternative recommendations in contrast with this? That would be very useful!
throwaway81523
Well, it depends on what you are looking for, but the traditional-style book that I'm used to is H. B. Enderton, "A Mathematical Introduction to Logic", and Enderton also has an accompanying book on set theory. I guess I'd suggest those as a companion rather than an alternative to the Gallier and Quaintance book. The G&Q book is imho more modern and more interesting from a CS-oriented perspective. It just leaves out too many important fundamentals. Alternatively, since it is still a work in progress, maybe some of the missing stuff could be added.
Boolos and Jeffrey "Logic and Computability", whatever the current edition is, is supposed to be good, but I haven't read it yet.
The G&Q book is weird in that it is about interesting topics that I'd consider slightly advanced, but written in a way that assumes almost no background, so if it's the only thing you read, you'll still be missing important background. I'd like a book that uses the G&Q approach but that includes the background. If it kept all the current contents and was expanded, it would probably end up as 2 volumes, which is fine. Or it could leave out some of the specialized stuff, but I think that stuff is there because that was what the authors wanted to write about, so I won't judge ;-).
throwaway81523
Can't edit to add this, so will post additional reply of an MO thread with a bunch of other recommendations. I will look into some of them myself.
https://mathoverflow.net/questions/44620/undergraduate-logic...
ProfHewitt
Unfortunately, the text by Gallier and Quaintancis is somewhat obsolete because it misses the following developments:
* The Church/Turing Thesis is false for reasons outlined here:
https://professorhewitt.blogspot.com/2021/03/the-churchturin...
* Gödel failed to prove inferential undecidablity in foundations for reasons outlined here:
https://professorhewitt.blogspot.com/2021/03/godel-failed-to...
* That "theorems of an order can be computationally enumerated" is true but unprovable for reasons outlined here:
https://professorhewitt.blogspot.com/2021/03/churchs-dilemma...
However, the text by by Gallier and Quaintancis does provide historical perspective.
drdeca
Let ◻ be the modal operator "It is provable that", so ◻P means that P is provable.
The first part of your argument in the second link says:[
Suppose that there is a proposition UNP such that UNP ⇔ ¬◻UNP .
Then, suppose that ¬UNP . It then follows that ¬¬◻UNP , i.e. that ◻UNP.
Therefore, ◻◻UNP .
Then, as UNP ⇔ ¬◻UNP, therefore ¬UNP ⇔ ◻UNP.
Then, replace the ◻UNP in ◻◻UNP with ¬UNP, and therefore conclude ◻¬UNP .
So, at this point, we have concluded that ◻¬UNP and also that ◻UNP. So, you can conclude that, __under these assumptions__, that ◻⊥ .
So, under the assumption that there is a statement UNP such that UNP ⇔ ¬◻UNP , and the additional assumption that ¬UNP, you can conclude that ◻⊥.
]
Indeed.
If a system admits a statement UNP, (and the system is strong enough to do the appropriate reasoning) (and peano arithmetic and such does admit such a statement UNP), then within the system, you can show that ¬UNP implies ◻⊥. I.e. "if the statement that claims it cannot be proven, can be proven, then the system proves a contradiction". This is entirely normal, and does not overturn anything.
However! While the assumption that ¬UNP allows us to derive ◻⊥, it does not allow us to derive ⊥. We have __not__ shown that ¬UNP → ⊥, we have only shown that ¬UNP → ◻⊥. Therefore, this is not justification in appealing to proof by contradiction, and concluding UNP (nor ◻UNP).
bloak
I've not read ProfHewitt's argument, so I'm just looking at your summary.
If ◻P means "P is a proposition that is formally provable within some implicit formal system" (perhaps there should be a subscripted letter to show which formal system because there are an infinite number of them), then I don't understand what ◻◻UNP means, because ◻UNP is not a proposition of the implicit formal system.
drdeca
If we were doing this in Peano Arithmetic using Gödel numberings, then ◻P would be expression by taking the Gödel number encoding P and applying a predicate that expresses provability to that number. This might be written as like, Provable«P» with the quotes indicating taking the Gödel number of the proposition P, in order to encode it in a way that Peano arithmetic can talk about. (Usually one uses the right angle brackets on the upper corners for the quoting, but my phone doesn’t have that on the keyboard, which is why I’m using “«“ and “»”). Statements like this are statements in the same formal system as P.
The modal logic known as “provability logic” is a nice abstracted system which captures how these things work, and allows for dealing with them in a general case.
◻◻P is perfectly sensible, and if the system is able to describe itself sufficiently well (which is not a high bar; Peano arithmetic satisfies this requirement.) then if you can derive P within the system (without additional assumptions), then, simply by including a proof that the proof of P is a proof of P, one can produce a proof in the system of ◻P, And so, if one can derive ◻P, so too can one derive ◻◻P .
(ProfHewitt’s post also uses basically the same thing as ◻◻P , though he writes it using the turnstile symbol rather than the square symbol, and I prefer the square symbol for it, and the turnstile for something slightly different. Not saying the use of the turnstile for it is wrong though.)
chriswarbo
> If ◻P means "P is a proposition that is formally provable within some implicit formal system" (perhaps there should be a subscripted letter to show which formal system because there are an infinite number of them)
It doesn't mean that. The formal system is explicit: it's the system we're using to write statements like `◻P` (it's a form of modal logic https://en.wikipedia.org/wiki/Modal_logic ).
ProfHewitt
Excellent question Bloak!
The proof is for a foundation that can formalize its own
provability. Consequently, ⊢⊢I’mUnprovable means that
⊢I’mUnprovable is provable in the foundation.
ProfHewitt
Prove I’mUnprovable using proof by contradictions as follows:
In order to obtain a contradiction, hypothesize
¬I’mUnprovable. Therefore ⊢I’mUnprovable
(using I’mUnprovable⇔⊬I’mUnprovable). Consequently,
⊢⊢I’mUnprovable using ByProvabilityOfProofs
{⊢∀[Ψ:Proposition<i>] (⊢Ψ)⇒⊢⊢Ψ}. However,
⊢¬I’mUnprovable (using I’mUnprovable ⇔⊬I’mUnprovable),
which is the desired contradiction in foundations.
a theorem using proof by contradiction in which
¬I’mUnprovable is hypothesized and a contradiction derived.
In your notation, the proof shows that following holds:
¬I’mUnprovable ⇒ ⊥
¬I’mUnprovable ⇒ ⊢⊥drdeca
Because ◻P and ◻¬P together imply ◻⊥, not ⊥.
Edit: if you had as an axiom, or could otherwise prove within the system, that ¬◻⊥, I.e. that the system is consistent, then you could conclude from ◻P and ◻¬P that ⊥, by first concluding ◻⊥, and then combining this with ¬◻⊥ .
And in this case, you could indeed say that this is a contradiction, and therefore reject the assumption of ¬UNK, and then conclude UNK without assumptions.
So, if one could show ¬◻⊥ (or if the system had it as an axiom), the reasoning would go through. (This is the thing that is missing.)
Therefore, you could then prove ◻UNK, and therefore ¬UNK, and therefore (having already shown UNK) would have a contradiction, ⊥.
So, this is a way of showing that, if you can show ¬◻⊥ (and if there is a statement UNK), then the system is inconsistent. Which is just one of Gödel’s theorems: a strong system can’t prove its own consistency without being inconsistent.
ganafagol
I'm having trouble understanding whether you are
(1) a troll hiding behind Carl Hewitts name using a veil of CS lingo that's just complex enough to appear legit but is actually nonsense, or
(2) actually Carl Hewitt losing his mind with a bunch of nonsense, or
(3) actually Carl Hewitt who is on to something big and the world has just not caught up with you.
To convince us of (3), have your theories been discussed at related conferences and what does the community make of them?
wildmanx
Carl has a long history of twisting truth and sources.
For example https://www.theguardian.com/technology/2007/dec/09/wikipedia...
Or the discussion page of the wikipedia page about him: https://en.wikipedia.org/wiki/Talk:Carl_Hewitt
Constantly using "Prof" to prefix his name speaks for itself too.
(In other words, it's (2), but that's not a recent development.)
amgreg
"Avoid unrelated controversies and generic tangents."
ProfHewitt
Unfortunately, Wikipedia has a long history of libeling people :-(
It would be great if HN could stick to substantive discussions
and not degrade into personal attacks.
ivanbakel
Having skimmed through the paper accessible at the end of the first link, I am leaning towards (1). It contains a barrage of eye-assaulting notation, lots of digressions and repetition, and basically no formal substance (which, for a logic/computation paper, is bizarre).
emmelaich
His WP talk page is a shit-show. I'm leaning towards 2.
https://en.wikipedia.org/wiki/Talk:Carl_Hewitt
https://en.wikipedia.org/wiki/Talk%3ACarl_Hewitt%2FArchive_2
(to disclose, I am not qualified to actually judge the arguments)
ProfHewitt
Sorry that mathematical notation is causing you problems.
How do you see the article lacking in formal substance?
amgreg
"Please respond to the strongest plausible interpretation of what someone says, not a weaker one that's easier to criticize. Assume good faith. "
https://news.ycombinator.com/newsguidelines.html
(1) does not assume good faith. (2) is not the strongest possible interpretation of what Hewitt said. Listing them here is flamebait and reflects poorly on the community. Shame.
Edit: Added guidelines link.
ProfHewitt
Dear ganafagol,
Your personal attacks should not be part of Hacker News discussions.
Please confine your remarks to substantive issues instead.
ganafagol
Um...? I was hoping to get some clarification that your account is legit and some references to opinions from the broader research community for your big claims. I came here with an open mind and would have liked to learn something since CS is my passion for decades now.
Unfortunately, by reading your comment history, I learned that you've been self promoting here for quite a while and have always been ignoring questions about references to other people's thoughts on your work. I also learned that there is a wikipedia talk page and apparently the same pattern happened there 5-6 years ago, where you were asked about citations and could only come with self-citation and claims of harrassment. And after sockpuppet edits you got banned there. That's sad, for everybody.
It would be so easy for you to just throw a handful of citations into the mix and let the research speak for itself! You want to go down in CS history as somebody having discovered something big, right? I'd love to see that! Then I could tell my grandkids that I was sitting in the front row! But you won't convince anybody by only doing self-promotion and complaining about "personal attacks". Scientific progress works by convincing the scientific community, and for that you need to engage with it positively.
dandanua
That book doesn't touch any nondeterministic behavior at all. So, the nondeterministic extension of the Church-Turing Thesis is absolutely irrelevant here. Your other cited "developments" seem senseful only in your personal universe.
arethuza
Yeah - its a long time since I studied this kind of stuff but I'm pretty sure the Church–Turing thesis doesn't say anything about non-determinism or not - being more relevant to complexity than computability?
dandanua
The original thesis concerns only computable functions, which are deterministic by definition. It doesn't even consider their complexity. There are extended variations of the thesis, though. The complexity variation means that any computable function in one model of computation can have only a polynomial slowdown factor in comparison with another model of computation. It's believed to be false since the raise of quantum computations.
ProfHewitt
The Church/Turing Thesis is that all computation can be
performed by a nondeterministic Turing Machine.
The Thesis is false because there are digital computations
that cannot be performed by a nondeterministic Turing Machine.
See the following article on how to formalize all of digital
computation:
"Physical Indeterminacy in Digital Computation"
undefined
ProfHewitt
Nondeterministic Turing Machines are the standard for the Church/Turing Thesis.
Do you have anything substantive to contribute to the discussion?
amgreg
I am disappointed at several comments on this thread. Professor Hewitt, I personally think you've enriched the community by putting these claims here. They've led me to read more about several things, which I will list here for anyone else who might be looking for related topics: The difference between indeterminism and nondeterminism, bounded and unbounded nondeterminism, the actor model of computation, local arbitration versus global consensus, and state machines versus configuration-based computation.
You've responded to the ad hominem attacks with class. I hope you do not take them as a reflection of the HN community at large; please keep coming back and posting (provocatively) as you have done, because it induces learning.
ProfHewitt
Thank you very much for your warm words.
Of course, I sometimes make mistakes :-(
However, I do try to learn from them!
ProfHewitt
PS. Also thanks for picking up on important topics including
"indeterminism and nondeterminism, bounded and unbounded
nondeterminism, the Actor model of computation, local
arbitration versus global consensus, and state machines
versus configuration-based computation."
jkhdigital
Ignore the downvotes: there is value in reading and attempting to understand some of this material, despite the inflammatory way the claims are made. Carl Hewitt is presumably in his 70s, so I'll forgive him for being an old man who probably doesn't give a shit what people think.
In particular, I find the discussion of "paradox" attacks to be quite illuminating. Bertrand Russell attempted to remove self-referential and self-applicable paradoxes through the embellishments of types and orders. Hewitt argues that Gödel basically "cheated" in his proofs because the Diagonalization Lemma violated restrictions on orders.
I'm not qualified to evaluate that argument on the basis of mathematical reasoning, but I do believe it points to a mismatch between the way computation is modeled in the abstract and the way it happens in reality. In the purely abstract world of natural numbers, there's really only one kind of thing: the natural number. Statements about natural numbers can be represented by natural numbers, statements about statements about natural numbers can also be represented by natural numbers, etc. While a given natural number can be parsed to compute if it is a valid encoding of a proposition of a certain order, it is always and forever a natural number and nothing else.
However, I'm not entirely convinced that this captures the nature of computation in reality. When computation is embodied in a physical system, is it possible that the output of some computation can itself manifest computational properties that are not mere compositions of the lower level computational model? Where the interactions between the "higher level" objects simply follow a different set of rules and thus form the foundation of a new formalism?
The notion of types seems to be a way to capture this, by introducing the possibility of distinguishing a "thing" from its abstract representation or description. If everything exists only within the world of the abstract representation, then a thing is indeed no different from its description, as any manipulation of the thing is equivalent to some manipulation of its description. But when the thing is, in fact, a physical object, it is clear that construction of the thing can fundamentally alter the model of computation. Why is this?
I suspect that it is because there are no "pure" or "inert" abstractions in the real world; everything is in fact performing computation all the time. So physical constructions are not just compositions of objects, but also compositions of the computational capabilities of those objects. I realize this probably sounds like navel-gazing at this point, but sometimes that can be a useful activity.
ProfHewitt
Thanks for your interesting ideas!
My main purpose is to educate. So I am pleased that you got something out of the article :-)
bmc7505
Why is this comment being downvoted?
exdsq
I assume because these are huge claims however they support these claims with their own blog site rather than a peer reviewed journal etc
ProfHewitt
The articles are published here:
https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=256...
drdeca
Because it is wrong. The comment is claiming that the comment author has overturned well understood results, when they have not done so.
ProfHewitt
You are welcome to join the debate!
Scientific development proceeds building on previous results
as is the case here as shown by references in articles.
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This is but one of many comprehensive texts about mathematics written Jean Gallier and Jocelyn Quaintance! They are all a little unpolished, but still an invaluable resource to students!
Here are some more book titles copied from his website: (https://www.cis.upenn.edu/~jean/home.html)
* Linear Algebra and Optimization with Applications to Machine Learning (html)
* Differential Geometry and Lie Groups (html)
* Homology, Cohomology, and Sheaf Cohomology (html)
* Proofs, Computability, Undecidability, Complexity, and the Lambda Calculus. An Introduction (pdf)
* Aspects of Harmonic Analysis and Representation Theory (html)
* Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning (html)
* Aspects of Convex Geometry Polyhedra, Linear Programming, Shellings, Voronoi Diagrams, Delaunay Triangulations (html)
* Notes on Primality Testing and Public Key Cryptography Part 1: Randomized Algorithms Miller-Rabin and Solovay-Strassen Tests (08/2017) (pdf)
* Spectral Graph Theory of Unsigned and Signed Graphs Applications to Graph Clustering: A Survey (12/2014) Posted to arXiv as paper cs.CV arXiv:1311.2492 (pdf)
* Introduction to discrete probability (12/2014) (pdf)