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The fall of the theorem economy

davidbessis.substack.com

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Re: The fall of the theorem economy

#121

Post by godelski »

observationist wrote: post #103
Terr_ wrote: post #94

Earlier quoted context omitted.

> there is no abstraction for which we lack the capacity to comprehend. How could this ever be tested/falsified? It feels a bit like "there is no idea we cannot think of." If we can't comprehend it, then it won't be an abstraction, it'll just be a mystery.

In principle - if you're able to scale appropriately, using technology to augment capacity, then in principle, there's no abstraction for which we lack the capacity to comprehend, because calculation is calculation. Turing Computers can calculate anything which can be calculated given enough time and memory. Brains are Turing complete. It's not just a tautology, it's a feature of the universe- if it can be computed,…

  > It's not just a tautology, it's
Inconsistent logic.

You've made a grave assumption in treating this as a binary problem: comprehensible or incomprehensible. The truth is that this lies along a spectrum. I can "comprehend" a color which my eyes can't see but this is different than trying to comprehend red. Or as a variant, I can comprehend a color that I can't see naturally but if I go into a lab and they simulate those cones in very specific ways I won't ever be able to really imagine it. Seeing it will be clearly a different level of comprehension.

  > In practice, we'll never run out of mystery or ignorance or mistakes.
I fully agree with this but also for some reasons unmentioned. Truth has infinite precision, a thing we will never achieve. Similarly my namesake showed there are limits to axiomatic systems.

  > Brains are Turing complete.
And there are problems that can't be solved by Turing complete machines. That only means they can solve computable problems but there's plenty that aren't. Very famously the halting problem isn't. And in the real world there's many problems which are intractable. And some are more intractable than others

Re: The fall of the theorem economy

#123

Post by lioeters »

pegasus wrote: post #117
skoodge wrote: post #97

Earlier quoted context omitted.

> There's also a huge issue with your use of the word subjective - math is objective. Proofs remain stable whether it's humans or any other system that does the processing. We test that objectivity by comparing the subjective readings from individual humans, and if the tests all return the same results, we can confidently say that the resulting proof is an objective fact about reality. Subjective fundamentally means…

> [...] claiming that math is producing objective facts ignores at least a few hundred years of philosophy of mathematics (if not more). Even practicing mathematicians like Chaitin have described math as being more about inventing than discovering. Chaitin's work (including in his own opinion) moved the philosophical needle on maths towards "discovery" and away from "invention".

Indeed, his philosophy on the nature of mathematics is closer to "discovery", as in physics, than "invention" as in arbitrary game of rules and symbols (Hilbert) or language game (Wittgenstein).

I forget who said, "If you keep asking why, eventually you end up in the mathematics department." Calling it Platonism is a misunderstanding of how physics (and chemistry, biology, and the rest) emerge from mathematical structures and relationships that are more fundamental.

Re: The fall of the theorem economy

#124

Post by Terr_ »

atq2119 wrote: post #102
Terr_ wrote: post #94

Earlier quoted context omitted.

> there is no abstraction for which we lack the capacity to comprehend. How could this ever be tested/falsified? It feels a bit like "there is no idea we cannot think of." If we can't comprehend it, then it won't be an abstraction, it'll just be a mystery.

It could potentially be falsified by an encounter with really weird aliens. But I find it quite plausible because it feels like it's fundamentally "just" a restatement or a minor variation of the Church-Turing thesis.

I see it as implausible, and I find myself thinking of Godel's incompleteness theorem(s).

Somewhere out there are "correct" thoughts we can't practically think without changing how our brains work.

Re: The fall of the theorem economy

#125

Post by grantcas »

observationist wrote: post #85
TheOtherHobbes wrote: post #74

Earlier quoted context omitted.

I'd be very surprised if there aren't huge areas of undiscovered math that can't be explained with either geometric or algebraic views. Math is entirely subjective. "Proof" essentially means "Other educated practitioners have the same experience when trying to understand this." The logical steps that proofs are built on all have that common foundation. Our concept of logic based on our subjective experience of "truth…

This is a serious misconception of human cognitive abilities. We have the ability to abstract generally - there is no abstraction for which we lack the capacity to comprehend. We regularly visualize, contextualize, and satisfactorily explain systems with dozens of dimensions. The fact that we cannot hold 4,5+ spatial dimensions in our imaginations sufficiently to develop an intuition for navigation in that space and…

[dead]

Re: The fall of the theorem economy

#126

Post by rramadass »

TimorousBestie wrote: post #67
rramadass wrote: post #39

Earlier quoted context omitted.

Right; this is my viewpoint too. All the "pure mathematicians" have a bleak future where AI can do all the puzzle solving better and faster. They existed in their own world elevating "theorem proving within a formal system" as the central aspect of "proper" mathematics and everything else as ancillary. It always felt wrong to me that while the scientific method iterated starting with the "real world" viz. Observe, Me…

> pure mathematicians lost themselves in the formalization of hypothesizing/modeling and thus lost touch with mapping it to reality. You’re describing a very small fragment of total current mathematical labor. Very few people work solely on “formalization” and even e.g. model theory or type theory have real consequences.

I meant "Formalization" in the larger sense of the word.

The Definition/Theorem/Proof (DTP) emphasis (to the exclusion of everything else) for axiomatization in mathematics arose in the late-19th century and solidified in the 20th century. Prior to this, while mathematicians always practiced rigour, they valued insight/intuition more for understanding.

An attempt was made to teach "Modern Math" in a "New Math" manner which though withdrawn, has left its detrimental mark on mathematics education to this day. See the "Reception" and "Legacy" sections at https://en.wikipedia.org/wiki/New_Math

Excerpts:

Physicist Richard Feynman argued, "first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material."

In a 1971 article, mathematician René Thom rejected the New Math as "a test of memory that poisons intelligence" because of its complete neglect of intuition.

Mathematician and historian of mathematics Morris Kline observed that it was "practically impossible" to learn new mathematical creations without first understanding the old ones, and that "abstraction is not the first stage, but the last stage, in a mathematical development."

Mathematician and author George F. Simmons wrote in the algebra section of his textbook Precalculus Mathematics in a Nutshell (1981) that by focusing on form rather than substance, the New Math produced students who had "heard of the commutative law, but did not know the multiplication table."

Mathematician Laurent Schwartz described the new reforms as "very poor" pedagogy. For him, "The goal of mathematics is not to prove rigorously things that everyone knows. Instead, the goal is to find rich results and then, in order to make sure they are true, to prove them."

See also;

Against mathematical proof - https://mathwithbaddrawings.com/2021/05/12/against-mathemati...

Proof and Understanding in Mathematical Practice by Danielle Macbeth - https://journals.openedition.org/philosophiascientiae/712?la...

Re: The fall of the theorem economy

#127

Post by rramadass »

cubefox wrote: post #58
rramadass wrote: post #39

Earlier quoted context omitted.

Right; this is my viewpoint too. All the "pure mathematicians" have a bleak future where AI can do all the puzzle solving better and faster. They existed in their own world elevating "theorem proving within a formal system" as the central aspect of "proper" mathematics and everything else as ancillary. It always felt wrong to me that while the scientific method iterated starting with the "real world" viz. Observe, Me…

Yes. Though even philosophy, which doesn't have the "real world" iteration that science does, arguably doesn't have the problems of pure mathematics. Pure mathematicians create ever more abstractions and get lost in solving puzzles on how these abstractions logically relate to each other. But since these abstractions don't have any relevance outside of pure mathematics, it's an entirely self-referential game, like ch…

https://news.ycombinator.com/item?id=48791433
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