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What does the end of mathematics look like?

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Re: What does the end of mathematics look like?

#61

Post by daveguy »

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.

The human brain runs on about 20 watts of power -- the entire body on 80 watts (at rest). Those numbers are at least good within an order of magnitude. The largest supercomputer consumes 29 megawatts of power for 1.7 exaflops. And the largest supercomputers are nowhere near the flexible generality of a human brain -- they're calculating FFTs for the test.

The amount of parallelism in the human brain is enormous. Not just each neuron, but each synapse has computational capacity. That means ~10^14 computational units or 100 trillion processing units -- on about 20 watts.

That doesn't even touch the bandwidth issues. Getting the sensory input in and out of the brain plus the bandwidth to get all of the processing signals between each neuron is at least another petabit per second. So, on bandwidth capacity alone we are 25+ years away (assuming the last 25 years of growth continues). And in humans that comes with 18 years of training at that massive bandwidth and computational power.

Also, we have no idea what a general intelligence algorithm looks like. We are just now getting multimodal LLMs.

From the computational/bandwidth perspective we are still 30 years from a computer being able to process The information a single human brain does, except while consuming 29+ megawatts of energy. If you had to feed a human 29 megawatts worth of power no business would be profitable. Humans wouldn't even survive.

Sorry, but the notion that we are close to AGI because we have good word predictors is fantasy. But, there will be some amazing natural language human-computer interface improvements over the next 10 years!

Re: What does the end of mathematics look like?

#62

Post by lo_zamoyski »

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…

A recurring problem I see is that people have absolutely drunk the koolaid of the utilitarian worldview. They've been marinating in it for so long, with no exposure to anything else beyond this parochial and base existence, that they have no idea they've been marinating in it. Everything is reduced to economic exchange. Everything is reduced to economic output. Learning is reduced to something that has value only if it is "marketable", only if it results in more units of toilet paper. The human being disappears from the picture as a mere instrument of the economy, and economy that exits for its own sake, that hovers over us like a god.

Given that kind of picture of reality, it is little wonder that AI seems like such a profound threat to so many people (putting aside for the moment the distinction between the aspirations of AI companies and the actual affordances it possesses). If being human is to be an economic instrument, then any AI that could eliminate the economic value of human beings is something akin to extinction. The god of economics has no further need of you. You may die now.

But this utilitarian view of the world reeks of nihilism. It is the world of total work, of work for work's sake. We never inquire about the ends that are the very reason for work in the first place. We never come to an understanding that economies exists for us, that we create them for mutual benefit. And we never seem to grasp that the economic part of human life is only part of human life, that it exists for the sake of those parts of life, the more important and most important parts of life, that are not a matter of economics. We have come to view life as meaningless, so we run into the embrace of the god of economics, losing ourselves in its endless churn, its immediate goals, truncating our minds so that we do not conceive of anything else, longing to escape the horror of the abyss that awaits us outside of its dreary confines...

The point of studying something in a theoretical capacity is to understand it, not to produce something of economic value. Each person must come into understanding from a state of not understanding. Homo economicus does not comprehend this. Homo economicus lives to eat and shit and cum and to accumulate things.

Re: What does the end of mathematics look like?

#63

Post by IsTom »

lblume wrote: post #51
IsTom wrote: post #50

Earlier quoted context omitted.

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?

It might be using some nonlocal context e.g. something similar was proved earlier in the proof and the reader is assumed to remember and/or generalize this or there's some assumptions that are stated only in the beginning. There's also probably a bit of skipped proofs of the kind "I'll be able to do this if pressed, but so will the reviewer".

Re: What does the end of mathematics look like?

#64

Post by mathgradthrow »

The terrifying thing about lean and machine learning is not the idea that we will train computers on human written proof, but that we won't have to. With the rules of chess easily computed, a computer can use the bellman equations and self play to learn a policy that is vastly superior to human play, at least on the critical path.

The state space of mathematics is pretty different from chess, but I think ultimately, mathematicians are just running something like A* on the space of propositions, with a custom heuristic that is learned by approximating the result of running A* with that heuristic. where your error is just the difference between the actual and predicted length of proof.

Re: What does the end of mathematics look like?

#65

Post by [deleted] »

rcxdude wrote: post #56
Chris2048 wrote: post #55

Earlier quoted context omitted.

> 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.

[deleted]

Re: What does the end of mathematics look like?

#66

Post by lioeters »

daveguy wrote: post #61
Chris2048 wrote: post #55

Earlier quoted context omitted.

> 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.

The human brain runs on about 20 watts of power -- the entire body on 80 watts (at rest). Those numbers are at least good within an order of magnitude. The largest supercomputer consumes 29 megawatts of power for 1.7 exaflops. And the largest supercomputers are nowhere near the flexible generality of a human brain -- they're calculating FFTs for the test. The amount of parallelism in the human brain is enormous. Not…

The next logical step, perhaps ethically questionable, seems to be growing human brains for computational purposes (parallel or quantum) with high bandwidth and very efficient power consumption.

Re: What does the end of mathematics look like?

#67

Post by Ukv »

rcxdude wrote: post #56
Chris2048 wrote: post #55

Earlier quoted context omitted.

> 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.

> while the silicon closest to its capabilities needs more like tens of kW.

I think looking at power consumption for the very edge of what technology is just barely capble of may be misleading, since that's inherently at one extreme of the current cost-capability trade-off curve[0] and stands to drop the most drastically from efficiency improvements.

You can now run models equivalent in capability to initial version of ChatGPT on sub-20w chips, for instance. Or, looking over a longer timeframe, we can now do far more on a 1-milliwatt chip[1] than on the 150kW ENIAC[2].

[0]: https://i.imgur.com/GydBGRG.png

[1]: https://spectrum.ieee.org/syntiant-chip-plays-doom

[2]: https://cse.engin.umich.edu/about/history/eniac-display/

Re: What does the end of mathematics look like?

#68

Post by daveguy »

lioeters wrote: post #66
daveguy wrote: post #61

Earlier quoted context omitted.

The human brain runs on about 20 watts of power -- the entire body on 80 watts (at rest). Those numbers are at least good within an order of magnitude. The largest supercomputer consumes 29 megawatts of power for 1.7 exaflops. And the largest supercomputers are nowhere near the flexible generality of a human brain -- they're calculating FFTs for the test. The amount of parallelism in the human brain is enormous. Not…

The next logical step, perhaps ethically questionable, seems to be growing human brains for computational purposes (parallel or quantum) with high bandwidth and very efficient power consumption.

Unfortunately we are much farther from growing a human brain than we are from a scaling up an LLM to a 29 megawatt-consuming behemoth.

With growing a brain, we barely know where to begin. Not in terms of growing a few neurons in a petri dish. Nourishing the complex interconnecting structure of neurons that is a human brain is nowhere even on the horizon. Much less growing the structure from cells. At least with the LLM/AI techniques we have control over the entire processing pipeline.

And I agree, that is an ethical minefield.

Re: What does the end of mathematics look like?

#69

Post by npodbielski »

lioeters wrote: post #66
daveguy wrote: post #61

Earlier quoted context omitted.

The human brain runs on about 20 watts of power -- the entire body on 80 watts (at rest). Those numbers are at least good within an order of magnitude. The largest supercomputer consumes 29 megawatts of power for 1.7 exaflops. And the largest supercomputers are nowhere near the flexible generality of a human brain -- they're calculating FFTs for the test. The amount of parallelism in the human brain is enormous. Not…

The next logical step, perhaps ethically questionable, seems to be growing human brains for computational purposes (parallel or quantum) with high bandwidth and very efficient power consumption.

I hope not. W40K is grim and dystopian for a reason and it have cogitors which is basically human-computers.

Re: What does the end of mathematics look like?

#70

Post by Chris2048 »

bubblyworld wrote: post #14
BSDobelix wrote: post #11

Earlier quoted context omitted.

>>I live by myself in a remote mountain cave = I live in California, and the nearest Starbucks is more than 20 miles away. >>can only be contacted during a full moon = As a night person, I am awake when the streetlight outside my house turns on. >>certain arcane rites that are too horrible to speak of = In order to contact me, you must install Microsoft Teams. Overall, it's not that bad, except for the MS team thing.…

The man is south african, we are geologically blessed and have a lot of pleasant remote mountain caves =)

The moon thing might still not be literal, similar to "once in a blue moon" i.e. rarely.
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