Hacker News

Arithmetic Without Numbers – How LLMs Do Math

85 points by old_sound ago | 26 comments

Npovview |next [-]

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).

xyzsparetimexyz |next |previous [-]

> 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!

stared |next |previous [-]

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.

0x59 |next |previous [-]

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 |root |parent [-]

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

iammjm |next |previous [-]

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

ACCount37 |root |parent |next [-]

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.

defrost |root |parent |next |previous [-]

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 |root |parent |next |previous [-]

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.

steveBK123 |root |parent [-]

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".

beernet |root |parent |next [-]

> while on the other hand, they are duct-taping in deterministic plugins for specific prompt types they find it better to offload

So, in essence, just like human beings?

tzs |root |parent |next |previous [-]

That doesn't seem very persuasive. The one example of a non-A GI we have, humans, does the same thing. We've been offloading arithmetic for at least 4000 years.

singpolyma3 |root |parent |previous [-]

> 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"

That is my hopeful ideal

steveBK123 |root |parent [-]

In which case it’s just a neat extension of search

breezybottom |root |parent |previous [-]

ChatGPT does, and has since 2023

euroderf |next |previous [-]

The spirit of Rube Goldberg is alive and well.

soupspaces |root |parent [-]

We evolved to do incremental fixes, not full refactoring

old_sound |next |previous [-]

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

rubyfan |next |previous [-]

Why does every exhibit made with AI look the same?

silvestrov |next |previous [-]

This is a very nice and fresh page layout.

dominotw |next |previous [-]

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.

singpolyma3 |root |parent [-]

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

andrewstuart |previous [-]

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

singpolyma3 |root |parent |next [-]

Usually yes

mavhc |root |parent |previous [-]

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 |root |parent [-]

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