Doing Mathematics Differently
inference-review.com
Doing Mathematics Differently
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Re: Doing Mathematics Differently
#2Somewhat related, a physics talk by Susskind about some interesting links between the inside of a black hole and (quantum) computational complexity. The complexity part starts at 20:00. He works with Scott Aaronson on this.
Re: Doing Mathematics Differently
#3Counterpoint (from my blog):
http://forwardscattering.org/post/7
http://forwardscattering.org/post/14
In summary i don't think AIT offers an absolute measure of complexity, due to having to choose the abstract machine.
Not to say the ideas aren't interesting, and that this isn't a nice article, from one of the main figures in the field.
Re: Doing Mathematics Differently
#4This is the Chaitin of Kolmogorov-Solomonoff-Chaitin complexity, if the argument seems familiar.
I like Grassberger-Crutchfield-Young (http://www.scholarpedia.org/article/Complexity, look for Statistical complexity) complexity, because it can actually be measured.
I had long suspicions that Chaitin was a Leibnitzian, but there you go. I like thinking about the Principle of Sufficient Reason, too, but I have long suspected the phenomenon of causality can be more simply explained by positive feedback effects _only_. (http://howonlee.github.io/2016/01/21/Poking-20At-20Causation...)
Re: Doing Mathematics Differently
#5Anyone fascinated by the (non-)intelligibility of the world might like one of Chomsky's many talks on the subject. Such as https://chomsky.info/20060301/
> In fact, if you look at the history of science seriously, in the seventeenth century there was a major challenge to the existing scientific approach. I mean, it was assumed by Galileo and Descartes and classical scientists that the world would be intelligible to us, that all we had to do was think about it and it would be intelligible.
> Newton disproved them. He showed that the world is not intelligible to us. Newton demonstrated that there are no machines, that there’s nothing mechanical in the sense in which it was assumed that the world was mechanical. He didn’t believe it — in fact he felt his work was an absurdity — but he proved it, and he spent the rest of his life trying to disprove it. And other scientists did later on. I mean, it’s often said that Newton got rid of the ghost in the machine, but it’s quite the opposite. Newton exorcised the machine. He left the ghost.
> And by the time that sank in, which was quite some time, it just changed the conception of science. Instead of trying to show that the world is intelligible to us, we recognized that it’s not intelligible to us. But we just say, ‘Well, you know, unfortunately that’s the way it works. I can’t understand it but that’s the way it works.’ And then the aim of science is reduced from trying to show that the world is intelligible to us, which it is not, to trying to show that there are theories of the world which are intelligible to us. That’s what science is: It’s the study of intelligible theories which give an explanation of some aspect of reality.
Re: Doing Mathematics Differently
#6"Gödel incompleteness is even unpopular among logicians. They are ambivalent. On the one hand, Gödel is the most famous logician ever. But, on the other hand, the incompleteness theorem says that logic is a failure."
I suspect Gödel himself didn't like the incompleteness for similar being very much an idealist who even created (but didn't) publish a modal-logic proof of the existence of God.
The thing is that if one takes a formalist position, that mathematics is a game played by pencil and paper (or computers) then the completeness and incompleteness theorems mean that foundational questions are simply done. oppositely, the position that foundations-people now have to take is that even though you create a universe compatible with any true-but-unprovable position (like the continuum hypothesis or its negation), some of these true-but-unprovable hypotheses are more plausible, aesthetically appealing or something and these hypotheses should be the one considered "true" in some ideal reality (I'm trying to crudely paraphrase Raymund Smullyan here).
Re: Doing Mathematics Differently
#7What's the point of inserting untranslated French text into an English article?
Is it supposed to just be window-dressing? Is it supposed to promote the (outdated and highly dubious) notion that all educated people speak French? Is it just for the author to show off? Whatever the reason, it's a highly obnoxious practice and it doesn't improve the article.
Re: Doing Mathematics Differently
#8This is the Chaitin of Kolmogorov-Solomonoff-Chaitin complexity, if the argument seems familiar. I like Grassberger-Crutchfield-Young ( http://www.scholarpedia.org/article/Complexity , look for Statistical complexity) complexity, because it can actually be measured. I had long suspicions that Chaitin was a Leibnitzian, but there you go. I like thinking about the Principle of Sufficient Reason, too, but I have long su…
The thing is that any computable complexity measure allows one to algorithmically produce an infinite which seems to have a complexity which goes to infinite as it get longer but since it is the product of a finite length computer program has finite complexity.
On the other hand, you can prove that for "nearly all" finite sequences of symbols, the algorithmic complexity is within a constant of naive statistical measures.
Re: Doing Mathematics Differently
#9This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics:
> the principle that mathematical truth is black or white and provides absolute certainty.
> Pure mathematicians like to think that they have absolute truth
Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an axiom reflects reality or not is out of scope, which is the whole point of their invention. As a philosopher you're welcome to debate them, and as a pragmatist you're welcome to choose them (and be compelled to accept their consequent theorems).
> Because it is easy for mathematicians to ignore Gödel’s proof. What lurks in their heart of hearts is a commitment to absolute truth, and a universal formal axiomatic theory for all of mathematics.
In one sense it's disheartening that we cannot prove or disprove every proposition in every axiomatic system. This is an unreasonable expectation as Gödel proved, but it doesn't invalidate the theorems we have proven and disproven. It also doesn't invalidate all of the programs we have written, even though there are non-terminating programs and uncomputable numbers.
Re: Doing Mathematics Differently
#10Counterpoint (from my blog): http://forwardscattering.org/post/7 http://forwardscattering.org/post/14 In summary i don't think AIT offers an absolute measure of complexity, due to having to choose the abstract machine. Not to say the ideas aren't interesting, and that this isn't a nice article, from one of the main figures in the field.
I don't think your claims are correct.
If you read a formal definition of Chaitin-Kolmogorov, you can find that the concept are essentially normalized by recursive function theory. By using the Church-Turing thesis, you can see that any program in any language can be simulated up to a constant length-multiple by an abstract Turing machine (speed isn't considered in these definitions so architecture doesn't matter much). Chaintin-Kolmogorov basically considers the log of the length of programs and then thus takes it's values as being plus or minus a constant for a given measure.