Arithmetic Without Numbers – How LLMs Do Math

https://alvaro-videla.com/llm-arithmetic-internals/article_interactive/article.html

by old_sound • 2 months ago

128 39 2 months ago

alvaro-videla.com

old_sound• 2 months ago

What happens inside an LLM when it tries to calculate with nothing but matrices.

silvestrov• 2 months ago

This is a very nice and fresh page layout.

euroderf• 2 months ago

The spirit of Rube Goldberg is alive and well.

soupspaces• 2 months ago

We evolved to do incremental fixes, not full refactoring

cwmoore• 2 months ago

Maybe evolutionary, but not ours, as the things we tend to want to refactor have come to exceed our lifespans.

iammjm• 2 months ago

Why doesn’t it just call tools such as Mathematica for such operations?

defrost• 2 months ago

This is more how an LLM thinks about math internally - an LLM version of drilled tables being used for mental arithmetic "as humans do".

When humans stall on these tasks, they reach for pen and paper, a slide rule, a calculator, etc.

Mathematica is overkill for arithmetic, in addition it's licenced and can cost a bit extra.

If an LLM were to reach for a light cheap arithmetic tool something like bc would be a good first stop - a CLI tool with a language that supports arbitrary precision numbers with interactive execution of statements.

https://en.wikipedia.org/wiki/Bc_(programming_language)

jampekka• 2 months ago

They do. I asked CharGPT for 327 x 48 and it used the "ChatGPT Instruments" calculator.

Previously it used to run Python scripts, and may still do for more complex calculations.

steveBK1234• 2 months ago

What's interesting is that one one hand LLM pumps are claiming a path to AGI.. while on the other hand, they are duct-taping in deterministic plugins for specific prompt types they find it better to offload...

In X years is it just going to be a thin OS-like layer where a majority of work is being handled by other "programs".

ACCount37• 2 months ago

For the same reason you don't run "4+6" on a calculator.

External tool call has an overhead. It requires a round trip into an external tool. It requires an LLM to run in agentic autoregression - it can't be used in prefill.

Which means that having native arithmetic capabilities is useful. Forward pass arithmetics are an LLM version of quick mental math.

An LLM can read "#define SILLY_TIME_CONST (3*20*60*60*1000)" and have "SILLY_TIME_CONST is 60 h expressed as 216000000 ms" already cached by the end of the line, before it even emits its first token.

breezybottom• 2 months ago

ChatGPT does, and has since 2023

ragebol• 2 months ago

I was thinking the same thing. Why not call into a dedicated math tool?

But I don't as well, and I have some intuition about numbers that I would probably not have if I always relied on calculators. Would the same sort of thing apply to LLMs? I'm probably anthropomorphising here...

andrewstuart• 2 months ago

I assumed it wrote Python or some sort of other code.

mavhc• 2 months ago

writing and calling an entire python setup seems massive overkill, surely just have an internal way of calling a simple calculator function would be millions of times faster

sebzim4500• 2 months ago

Probably but the cost of running a short lived python interpreter to run "print (100 + 200)" is likely negligable compared to the cost of running the language model itself

singpolyma33• 2 months ago

Usually yes

Npovview• 2 months ago

Turing Award Winner: Thinking Clearly, Paxos vs Raft, Working With Dijkstra | Leslie Lamport

https://www.youtube.com/watch?v=U719vQz-WFs

Leslie Lamport : "I am not smart. I have the gift of abstraction."

Real mathematics isn't about details. Its about concepts and abstractions and how we compose them (LLMs are good at those aspects).

helterskelter• 1 month ago

That's an interesting quote, because Feynman's superpower seemed to be his ability to visualize a difficult problem and make it parsable by mere mortals. I think he only scored ~135 on an IQ test (whatever that's worth).

BobbyTables24• 1 month ago

Pity Feynman didn’t write a Distributed Systems textbook…

stared• 2 months ago

There is a beautiful MathOverflow thread on how mathematicians imagine concepts, https://mathoverflow.net/questions/38639/thinking-and-explai....

Very often it involves spatial thinking. Vide one example there:

Once I mentioned this phenomenon to Andy Gleason; he immediately responded that when he taught algebra courses, if he was discussing cyclic subgroups of a group, he had a mental image of group elements breaking into a formation organized into circular groups. He said that 'we' never would say anything like that to the students. His words made a vivid picture in my head, because it fit with how I thought about groups. I was reminded of my long struggle as a student, trying to attach meaning to 'group', rather than just a collection of symbols, words, definitions, theorems and proofs that I read in a textbook.

stymaar• 2 months ago

There is a beautiful MathOverflow thread on how mathematicians imagine concepts, https://mathoverflow.net/questions/38639/thinking-and-explai....

And obviously Terrence Tao is up there in the response.

dominotw• 2 months ago

i dont like this new trend of generating html with ai to say something. i think some guy from anthropic started this trend .

now everything looks the same and i can no longer read on kindle.

singpolyma33• 2 months ago

Everything looked the same before too. One of the same 6 Jekyll temples etc. Fads in design come and go

rubyfan• 2 months ago

Why does every exhibit made with AI look the same?

0x59• 2 months ago

One could use many things to do arithmetic:

- color wheel

- oxidation reactions

- interpretive dance

- migratory patterns of curlew sandpipers

Whether one should is another question

throw1234567891• 2 months ago

“You know how when you see prime numbers, they appear red, but when they're twin primes, they're pink and smell like gasoline?”

xyzsparetimexyz• 2 months ago

The original dream A just-in-time compiler for arithmetic

What is it with LLM writing where it gives a smaller heading just before the main heading? Its nonsensical!

zdc1• 2 months ago

You'd think with tool use being as available as it is, the first tool we'd want to give them is a calculator...

Arithmetic Without Numbers – How LLMs Do Math

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