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What does the end of mathematics look like?

awanderingmind.blog

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Re: What does the end of mathematics look like?

#51

Post by lblume »

IsTom wrote: post #50
lblume wrote: post #35

Earlier quoted context omitted.

> requires knowledge of the proof itself (in general) Why? If a proof is wrong it has to be locally invalid, i.e. draw some inference which is invalid according to rules of logic. Of course the antecedent could have been defined pages earlier, but in and of itself the error must be local, right?

Human-written proofs are not written in Lean to be checked easily and there'll be potentially many formalizations for written prose and only some of them will be what the author intended. You need to pick the right formalization before you can say that this proof has local errors.

If a human reviewer rejects a proof, what do they consider apart from local errors? Formal proofs have formal local errors (which can be checked mechanically), while informal proofs have informal local errors (which require humans and some degree of hermeneutics to check). Am I missing something?

Re: What does the end of mathematics look like?

#52

Post by im3w1l »

Someone wrote: post #33
hliyan wrote: post #5

Considering that mathematics is, at its core, a language for defining relationships between quantities, and then relationships between those relationships, so on and so forth, I think it's fair to assume that the possible number of such relationships are infinite. Some of these relationships will obviously be useful in the real world, but they don't always have to be. I too, suspect that we can keep on building theor…

> I think it's fair to assume that the possible number of such relationships are infinite That’s easily proven to be true. “Two plus two equals four” is a theorem, so is “three plus three equals six”, etc.

Well I think the question should be how many interesting relationships there are.

Re: What does the end of mathematics look like?

#53

Post by [deleted] »

prennert wrote: post #45
jcelerier wrote: post #40

Earlier quoted context omitted.

> The camera didn't kill painting But it did. Painter used to be a trade where you could sell your painting skills as, well, a skill applicable for other than purely aesthetic reasons, simply because there were no other ways to document the world around you. It just isn't anymore because of cameras. Professional oil portrait painter isn't a career in 2025.

The Royal Society of Portrait Painters might disagree: https://therp.co.uk/artists/

[deleted]

Re: What does the end of mathematics look like?

#54

Post by ninininino »

awanderingmind wrote: post #42
dsign wrote: post #37

I love the language of this article :-)... it may be florid, but that's quintessentially human. About the substance, I agree that there are fair grounds for concern, and it's not just about mathematics. The best case scenario is rejection and prohibition of uses of AI that fundamentally threaten human autonomy. It is theoretically possible to do so, but since capital and power are pro-AI[^1], getting there requires a…

Thanks for the positive feedback on my writing style! Based on feedback in this thread it seems to be a divisive topic, haha.

I really appreciated it too.

Re: What does the end of mathematics look like?

#55

Post by Chris2048 »

npodbielski wrote: post #49

I think people like author are positive about us, humanity, being able to build AI or something being very close to that. I am not. From the energy efficiency perspective human brain is very, very effective computational machine. Computers are not. Thinking about scale of infrastructure of network of computers being able to achieve similar capabilities and its energy consumption... it would be enormous. With big infr…

> he energy efficiency perspective human brain is very, very effective computational machine

Can you explain why you think that? Very often, mechanical efficiency outperforms biological. Humans have existed for thougsands of years, neurons even longer. Computers and AI and relatively recent, we haven't really begun to explore optimisation possibilities.

Re: What does the end of mathematics look like?

#56

Post by rcxdude »

Chris2048 wrote: post #55
npodbielski wrote: post #49

I think people like author are positive about us, humanity, being able to build AI or something being very close to that. I am not. From the energy efficiency perspective human brain is very, very effective computational machine. Computers are not. Thinking about scale of infrastructure of network of computers being able to achieve similar capabilities and its energy consumption... it would be enormous. With big infr…

> he energy efficiency perspective human brain is very, very effective computational machine Can you explain why you think that? Very often, mechanical efficiency outperforms biological. Humans have existed for thougsands of years, neurons even longer. Computers and AI and relatively recent, we haven't really begun to explore optimisation possibilities.

It may be possible to optimise silicon further, but the brain does all of its work with less than a hundred watts, while the silicon closest to its capabilities needs more like tens of kW.

Re: What does the end of mathematics look like?

#57

Post by Chris2048 »

lmm wrote: post #38

The camera didn't kill painting. Neither the bicycle nor the motor-car killed running. There are already subfields of mathematics where it's believed that all the interesting discoveries have been found and no-one is looking except for the occasional amateur - and other subfields where to even have a hope of doing cutting edge research you would need to both do multiple years of postgraduate study and then get accept…

> nor the motor-car killed running

Is running an art/vocation comparable to photography and/or painting? We no longer have mailmen who run the length of the country afaik.

But running did heavily contribute to sedentary lifestyles in western countries, along with a bunch of other things.

> mathematical research isn't even a significant employment area

I agree, I think it will move from mathematicians "doing" math, to managing computerised system that do it instead. I'm sure we already have such systems.

I think far more important to humanity is improving mathemetical-literacy. From my perspective, math is made for mathematicians - it could be more accesible. As "pure" amth matures, there is still plenty opportunity in "applied" math (however you might define it).

Re: What does the end of mathematics look like?

#58

Post by awanderingmind »

npodbielski wrote: post #46
awanderingmind wrote: post #42

Earlier quoted context omitted.

Thanks for the positive feedback on my writing style! Based on feedback in this thread it seems to be a divisive topic, haha.

I would say that I am envy that someone can write like that. I can't write in such manner in my native language, let alone in the second one: English. It is nice to read or hear someone speaking like that, considering we are surrounded by low quality, easy to consume content nowadays. And I am envy of such skill because I like to think about myself as not entirely being stupid, still I would never be able to write/sp…

English is my second language too - ironically I am not so as proficient in my mother tongue, due to globalisation/imperialism. But it's ok, fortunately I love the language!

Re: What does the end of mathematics look like?

#59

Post by bananaflag »

initramfs2 wrote: post #30
hliyan wrote: post #5

Considering that mathematics is, at its core, a language for defining relationships between quantities, and then relationships between those relationships, so on and so forth, I think it's fair to assume that the possible number of such relationships are infinite. Some of these relationships will obviously be useful in the real world, but they don't always have to be. I too, suspect that we can keep on building theor…

just the zfc axioms alone are already infinite. It's an axiom schema ranging over an infinite number of actual statements. That's just statements, without even considering symbols as you're saying.

Well you can just put the nbg instead, which is finitely axiomatizable

Re: What does the end of mathematics look like?

#60

Post by Chris2048 »

hliyan wrote: post #5

Considering that mathematics is, at its core, a language for defining relationships between quantities, and then relationships between those relationships, so on and so forth, I think it's fair to assume that the possible number of such relationships are infinite. Some of these relationships will obviously be useful in the real world, but they don't always have to be. I too, suspect that we can keep on building theor…

> the possible number of such relationships are infinite

I think you need to be careful taking about "infinite" in the context of math. If the number of quantities, relationships etc is finite, so are all their combinations. Even things like the infinit-ude of available numbers might have fixed patterns that render their relevant properties effecively finite, and lead to further distinctions e.g finite vs countable, etc.

Personally, I feel like math has a bit of a legacy problem. It holds on to the conventions of an art that is very old, with very different initial assumptions at its conception, and this is now holding it back somehow. I lack the background to effectivly demonstrate this other than "Things I know/understand seem less intutive in standard mathenatical terms" e.g. generating functions and/or integrals feel easier to understand (to me) when you understand the, to be software-like 'loops'.

In fact, the idea of "constructivist math" seems (again, to me) to beg for a more algorithmic/computational approach.

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