Arithmetic Without Numbers – How LLMs Do Math
alvaro-videla.comTuring 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).
> 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!
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.
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
“You know how when you see prime numbers, they appear red, but when they're twin primes, they're pink and smell like gasoline?”
Why doesn’t it just call tools such as Mathematica for such operations?
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.
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.
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.
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".
> 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?
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.
> 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
In which case it’s just a neat extension of search
ChatGPT does, and has since 2023
The spirit of Rube Goldberg is alive and well.
We evolved to do incremental fixes, not full refactoring
What happens inside an LLM when it tries to calculate with nothing but matrices.
Why does every exhibit made with AI look the same?
This is a very nice and fresh page layout.
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.
Everything looked the same before too. One of the same 6 Jekyll temples etc. Fads in design come and go
I assumed it wrote Python or some sort of other code.
Usually yes
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
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