Arithmetic Without Numbers – How LLMs Do Math

> 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.

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).

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.

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

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.

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...

ChatGPT does, and has since 2023

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

We evolved to do incremental fixes, not full refactoring

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

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

Usually yes