41–50 of 108 posts
I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs. I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adop…
This doesn't seem like a very good point to make, sure reals are uncountable and any set of them with labels is countable and of measure zero. That doesn't say anything about physics at all. In QM things are only discrete in certain ways, like energy levels, not positions. A wave function over space can take on any real valued value in its range. Probabilities are real numbers(norms of the wave functions), and there…
If any are a "randomly" chosen real number, the answer would almost certainly be no. But a test of sufficient precision would of course be impossible.
I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-phy…
Interesting, I think everyone else calls this Kolmogorov complexity.
Earlier quoted context omitted.
One could make the argument that the only numbers that actually “exist” are the natural numbers, but the question ultimately is can you model any real number in the physical universe. Modeling ½ is simply a question of picking a unit to be 1 and finding its midpoint (or for that matter, declaring two apples to be “1” and thus a single apple would be “½”, although it’s a bit of a challenge to use apples to model (2-√3…
Make a line of 10 apples and declare it to be 2 units long, then make a square that's 15 applies diagonal, finally measure how much longer the 10 apples are than the side of the square. It's a pretty linear increase in complexity between the math and the apples.
I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs. I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adop…
I think footnote 16 on pg 12 clarifies his view.
> Why should we believe in real numbers, if most of them, it turns out,[^15] are maximally unknowable like Ω? [^16]
The footnotes:
> [^15]: See the chapter entitled The Labyrinth of the Continuum in [Chaitin, 2005]
> [^16]: In spite of the fact that most individual real numbers will forever escape us, the notion of an arbitrary real has beautiful mathematical properties and is a concept that helps us to organize and understand the real world. Individual concepts in a theory do not need to have concrete meaning on their own; it is enough if the theory as a whole can be compared with the results of experiments.
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The reference [Chaitin, 2005] in footnote 15 links to..
Meta Math! The Quest for Omega - http://arxiv.org/abs/math/0404335
> This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications.
Irreducible Complexity in Pure Mathematics - http://arxiv.org/abs/math/0411091
> By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for all of mathematics.
Norman Wildberger is a required mention on this topic. Here's a great discussion on Curt Jaimungal's podcast: https://www.youtube.com/watch?v=l7LvgvunVCM And a good debate on the topic with Daniel Rubin, who takes the more orthodox position: https://www.youtube.com/watch?v=edh5bbgSKqo Wildberger has tons more on this topic on his own channel. His arguments are thought-provoking, even if you don't agree with them.
Earlier quoted context omitted.
Make a line of 10 apples and declare it to be 2 units long, then make a square that's 15 applies diagonal, finally measure how much longer the 10 apples are than the side of the square. It's a pretty linear increase in complexity between the math and the apples.
in what way does drawing a diagram have anything to do with physically realizing a number?
And I'm just explaining how to use the same methods as the comment I replied to.
Earlier quoted context omitted.
> What do "real" numbers buy you? They're well-known and have a simpler implementation, and we are familiar with their quirks. There is a giant body of useful knowledge built up around standard real analysis. That doesn't really exist if you insist on using only computable numbers. The computables are also more fiddly in many ways. Because equality is undecidable, you can't have discontinuous functions, you need to c…
> equality is undecidable equality is always undecidable until you see the light of intuition. consider the rational number whose numerator is 0 if $theorem is true, and 1 if it is false, and whose denominator is 1.
But there are numbers in constructivism for which it is unknown whether they are zero. Some of which must remain unknown, if mathematics is consistent. This is a rather important and weird edge case.
I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist. But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Simil…
>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? I guess the naive answer is completeness. Every Cauchy sequence converges to a member of the space. For example, quantum mechanics relies on the formalism of Hilbert space, defined as a complete inner product space. This gives us nice things like the spectral theorem for unbound…
What you get for the formalism around computable numbers is this. Every mathematical object in the theory is something that can be, at least in principle, actually written down. When we say that it exists, this existence is of the most tangible form that any mathematical thing could have.