The constructive reals aren't complete because there are too few of them, only countably many — fishfry
Here's my definition of infinity, and for simplicity I'm only referring to positive infinity: infinity is a number, but it has a characteristic that all real numbers do not possess. — Michael Lee
Good luck reading. MO as you know is a site for professional mathematicians so the best one can hope for is to understand a few of the words on the page. — fishfry
The standard reals are the Goldilocks model of the reals. Not too small and not too big to be Cauchy-complete. They're just right. And are therefore to be taken as the morally correct model of the reals. — fishfry
The constructive reals aren't complete because there are too few of them, only countably many
— fishfry
Too few...or too many? The subset of computable total functions that correspond to the provably convergent Cauchy sequences form a countable and complete ordered field, that is a proper subset of the provably total functions. — sime
My current understanding is that there exists indeed a detailed description of the infinite model(s) for real numbers but at this point I am unable to pierce through the dense vocabulary and concepts in order to develop a correct mental picture on the matter. — alcontali
I shudder to think of what would happen here if the posters on this and other threads with minimal mathematical knowledge apart from set theory and logic were to launch investigations into subjects like functional integration or even metric spaces or advanced calculus. — jgill
Curating the math corpus. So how big is the historical corpus of mathematics? There’ve probably been about 3 million mathematical papers published altogether—or about 100 million pages, growing at a rate of about 2 million pages per year. And in all of these papers, perhaps 5 million distinct theorems have been formally stated. — Stephen Wolfram on 'Curating the math corpus'
He attended the University of Waterloo but dropped out in 2014, when he received the Thiel Fellowship in the amount of $100,000,[10] and went to work on Ethereum full-time.[10] — Wikipedia on Vitalik Buterin
This is the third part of a series of articles explaining how the technology behind zk-SNARKs works; the previous articles on quadratic arithmetic programs and elliptic curve pairings are required reading, and this article will assume knowledge of both concepts. — Vitalik Buterin's article intro
Talk is cheap. Show me the code. — Linus Torvalds on 'cheap talk'
I shudder to think of what would happen here if the posters on this and other threads with minimal mathematical knowledge apart from set theory and logic were to launch investigations into subjects like functional integration or even metric spaces or advanced calculus. — jgill
But maybe there is a hidden reservoir of mathematical understanding just waiting for opportunities for expression. I may try starting a thread and see what happens. I know several of you have significant mathematical depth. But others? Not so sure. :smile: — jgill
Too few, clearly. There are only countably many of them.
...
And no countable ordered field can be complete. It's a theorem. — fishfry
The computable total functions are sub-countable. An enumeration of all and only the constructively convergent cauchy sequences isn't possible as this is equivalent to deciding every mathematics proposition. Nevertheless we can construct a countable enumeration of a proper subset of the computable total functions, namely the provably convergent cauchy sequences with locateable limits, which collectively constitute a complete and ordered field, where by "complete" we mean with respect to a constructive least upper-bound principle. — sime
Too many Wiki pages, not enough math, that's my diagnosis of your posts. — fishfry
The computable numbers are countable. That's because the set of Turing machines is countable. Over a countable alphabet there are countably many TMs of length 1, countably many of length 2, etc.; and the union of countable sets is countable. QE Freaking D. — fishfry
The sequence of n-th truncations of the binary expansion of Chaitin's number is a Cauchy sequence that does not converge to a computable real. End of story. Then you say, "Oh but that sequence isn't computable," and I say, "So freaking what?" and this goes on till I get tired of talking to yet another disingenuous faux-constructivist. — fishfry
I cannot think of anything in mathematics or logic that is not a concept. — Michael Lee
Now that's something I've never run across. Both too big and too small at the same time. But it takes a weak form of the axiom of choice to have a nonprincipal ultrafilter, which is needed to construct the hyperreals. Do constructivists allow that? — fishfry
emmm......... Nope :) for the reason you've just mentioned. For where is the algorithm of construction? Of course , the trivial principle ultrafilter is permitted, which then produces a countable model.. — sime
By "constructive hyperreal" i was merely colloquially referring to using functions such as f(n)=1/n as numbers according to some constructive term-oriented method that didn't involve assuming or using cauchy limits. — sime
Just a guess, but I would imagine that one typically becomes a constructivist in the first place for primarily philosophical reasons--e.g., dissatisfaction with the philosophical basis of standard math, hence the desire for and advocacy of alternative definitions. Since this is a philosophical forum, rather than a mathematical forum, it is a natural place for committed constructivists to make their case.Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler. — fishfry
Wait so you just made that up? It's not a real thing? You had me convinced. Why not mod out the reals by the trivial ultrafilter and see what you get? What do you get?
Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler. — fishfry
Thanks for your post regarding mine, fishfry. Your quote above to sime is germane.
I’ve ruffled some feathers with my post, for which I apologize. I got a bit irritated last night and didn’t express my thoughts well. — jgill
First, I’m not coming from a feeling of superiority regarding math. As a retired prof my interests are in a sliver so small it’s barely visible, one low-interest page among 40,000 on Wikipedia. There are sophisticated discussions on this forum about math, computer science, and logic that I can only stand aside and watch. And most conversations about foundations are beyond me. — jgill
But sometimes posters will make statements about mathematics in general that are erroneous, but said with conviction. — jgill
Such as claiming that math proofs are computer programs, — jgill
or that there are no more geometrical proofs. — jgill
Or saying that fiddling with axioms makes the entire body of mathematics flawed, when, in fact, most mathematicians wouldn’t even notice. — jgill
Claiming that irrational numbers are a mistake and that this undercuts the entire structure of mathematics. — jgill
Stating that calculus is largely manipulating symbols — jgill
and that formal education is detrimental. — jgill
That adding a symbol, a “number”, for infinity will undermine current mathematics. — jgill
For misusing the expression “chaos theory” when discussing randomness. — jgill
For claiming that much of what we know of math now was derived or discovered two thousand years ago. — jgill
On and on. I've probably misinterpreted some of this. If so, apologies. — jgill
It’s this moving away from what one knows to speculative territory, but being convinced one is correct – that’s a little annoying to me. But this is a philosophy forum, so no harm done. — jgill
As for physics, well all is not well in that discipline. For example, there is an argument about the aether that seemingly goes as follows: The premise is that every wave must travel through a physical substance, and that the aether exists. Electromagnetic pulses are waves, therefore must be propagated through the aether. Hence, electromagnetic waves travel through a physical substance. Makes sense if the premise is true. It's conjecture stated as fact. — jgill
I took a year of physics in college, and as a math prof used some physics in my classes. But I would feel incompetent to engage in a discussion about anything beyond the simplest ideas. But here we have string theory, differentiable manifolds, general relativity, entanglement, Bell’s theorem, and on and on – all as if the poster is sure of what he is talking about and not merely parroting Wikipedia. Maybe it’s no more than a lack of modesty. If I have offended anyone, sorry. — jgill
Our friend Metaphysician Undercover is a special case. — fishfry
Thanks for the compliments. The biggest stumbling block between us is your concept of "mathematical existence". The proof that something has mathematical existence is really meaningless unless we have a rigorous definition, or convention, concerning what "mathematical existence" means. — Metaphysician Undercover
Such as claiming that math proofs are computer programs, — jgill
This is in fact true. It's the famous Curry-Howard correspondence — fishfry
Stating that calculus is largely manipulating symbols — jgill
To be fair, that's exactly how we teach it. — fishfry
and that formal education is detrimental. — jgill
Also to be fair, many of the high and mighty in the land say the same. — fishfry
You do give the impression of not having been on the Internet much — fishfry
Does the knight's move have chess existence? The other day you said you reject chess because it doesn't refer to anything in the real world. That's extreme nihilism. You can't get out of bed in the morning with a philosophy like that. How do you know it's your own bed? Property's an abstraction. — fishfry
The one I particularly enjoyed delved into the nature of mind and consciousness. — jgill
The proof that something has mathematical existence is really meaningless unless we have a rigorous definition, or convention, concerning what "mathematical existence" means. — Metaphysician Undercover
What could you possibly mean by "chess existence"? — Metaphysician Undercover
This is certainly valid regarding the structure of a mathematical argument. But by itself it leaves the impression that mathematics is merely symbol manipulation and not what it really is: exercises in imagination and creativity. — jgill
At the University of Chicago in the fall of 1958, I was surprised to learn that the physics department was no longer allowing its students to enroll in courses from the math department and was teaching its own mathematics. — jgill
Also, even though avatars were used we all knew the identities of the primary contributors. — jgill
Well obviously from a pure mathematics perspective, every proof in ZFC is considered construction, — sime
in contrast to Computer Science that has traditionally had more natural affinity with ZF for obvious reasons, and there is a long historical precedent for using classical logic and mathematics. — sime
As a language, there is nothing of course that classical logic cannot express in virtue of being a "superset" of intuitionistic logic, but classical mathematics founded upon classical set theory IS a problem, because it is less useful, is intuitively confusing, false or contradictory, lacks clarity and encourages software bugs. — sime
In my opinion, Constructive mathematics founded upon intuitionistic logic is going to become mainstream, thanks to it's relatively recent exposition by Errett Bishop and the Russian school of recursive mathematics. Constructive mathematics is practically more useful and less confusing for students in the long term. Consider the fact that the standard 'fiction' of classical real analysis doesn't prepare an engineering student for working in industry where he must work with numerical computing and deal with numerical underflow. — sime
The original programme of Intuitionism on the other hand (which considers choice-sequences created by the free-willed subject to be the foundation of logic, rather than vice versa) doesn't seem to have developed at the same rate as the constructive programme it inspired. However, it's philosophically interesting imo, and might eventually find an applied niche somewhere, perhaps in communication theory or game theory. — sime
BTW, i'm not actually a constructivist in the philosophical sense, since the constructive notion of a logical quantifier is too restrictive. In a real computer program, the witness to a logical quantifier isn't always an internally constructed object, but an external event the program receives on a port that it is listening. — sime
What's really needed is a logic with game semantics. Linear logic, which subsumes intuitionistic and classical logic is the clearest system i know of for expressing their distinction and their relation to games. — sime
As for a trivial ultrafilter, its an interesting question. Perhaps a natural equivalence class of Turing Machine 'numbers' is in terms of their relative halting times. Although we already know that whatever reals we construct, they will be countable from "outside" the model, and will appear uncountable from "inside" the model. — sime
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