July 18, 2023 · 5 mins · 1069 words

This is an old note I took to myself a few months ago when I was learning about computability theory. It was just a couple of bullet points, so I rewrote them in a better format and added some images. Hope you enjoy it.

During these last weeks I’ve been reading about computability theory. Of all the amazing things I learned, the ones that surprised me more are the ones related with the Halting Problem and Godel’s Incompleteness theorems. The ideas that the almighty maths have some limitations just made my head explode. I mean, I’ve been making heavy usage of maths all my life, first during my Physics degree and after while doing Machine Learning, and they never failed me - usually it was me who failed maths. So just the idea that maths can have some limitation was completely new for me. This first idea that math have some intrinsic limitations - eg: you can’t compute all the Busy Beaver Numbers - make me start wondering if there are other factors that limit maths. In this post I’ll write down the some random ideas I got from thinking about this topic.

# Maths are limited by maths

As I said in the introduction, there are some intrinsic limitations of maths. In some of my posts I’ve been talking about the Halting Problem and numbers that can’t be computed where it’s shown that not everything is possible in maths.

# Maths are limited by physics

In the last section, we saw that maths are not as omnipotent as one could though. However, we can go further and say that maths are also limited by the universe where we’re living. The argument goes as follows

1. Our brain is made of matter.
2. Matter follows physical laws.
3. Then, our brain follows physical laws.
4. We use our brain to generate mathematical ideas.
5. Then, the ideas we can generate are limited by physical laws.

Basically, this means that all the mathematical knowledge we have nowadays has been generated by using our limited-by-physics brain, so all this knowledge should be compatible with what our limited-by-physics brain can do, ie: maths are limited by the laws of our universe. Even mathematical ideas generated by computer do need to follow physical laws since computers (even quantum ones) do follow the laws of our universe.

# Maths are limited by biology

I’ve already made two hot takes, so why stop here? Let’s go even further and make all the mathematical community hate me. So far we’ve seen that since maths need a physical substrate to exist then they should follow physical laws. On the same lines, we can say that maths need a biological substract to be generated, so the ideas we can produce are limited by how humanity have evolved. One could argue that the current neural processes that our brain can perform are the ones that constrain the space of possible ideas.

Another limitation that biology poses to maths is that we only live for a limited number of years. So, at some point, the amount of time you would need to reach the boundaries of math knowledge is going to be larger that the human lifespan.

# Final thoughts

Let’s stop here since I don’t want to wake up tomorrow and find a mob of mathematicians outside my house ready to burn me in the town square. Here are some final thoughts about this topic.

First of all, at some point while writing this post I wrote the following line

There’s no pure knowledge, since the substract where the knowledge lies is material and thence not pure.

and suddenly I realized that someone else already said that a couple of years ago.

Even if what I’ve just said has been already known for centuries it’s nice to see that my random ramblings are not that absurd. However, I’m aware that the topic of math limitations has been probably studied much more in depth -and with much more rigor- and that what I’ve just said is probably wrong. If you have some resource about this topic I would be grateful to read it.

Also, after writing this text I asked myself “ok, so what?” Do we care about maths that we can’t even comprehend? As I read somewhere you don’t know what you don’t know, although in this case it should be you don’t know what you can’t know. Maybe we just care about exploring ideas within the boundaries defined by our limited existence.

Furthermore, another thing I noticed is that to show that my theory is right I need to present an idea that can’t be generated by the laws of the universe, but by definition this is impossible since AFAIK I’m still in this universe. So, does the question “are there ideas that can’t be generated in the universe?” even matter?

Let me finish with a quote by Scott Aaronson that exemplifies very well how our limitations shape the world we see the world and the way we create ideas

Indeed, one could define science as reason’s attempt to compensate for our inability to perceive big numbers. If we could run at 280,000,000 meters per second, there’d be no need for a special theory of relativity: it’d be obvious to everyone that the faster we go, the heavier and squatter we get, and the faster time elapses in the rest of the world. If we could live for 70,000,000 years, there’d be no theory of evolution, and certainly no creationism: we could watch speciation and adaptation with our eyes, instead of painstakingly reconstructing events from fossils and DNA. If we could bake bread at 20,000,000 degrees Kelvin, nuclear fusion would be not the esoteric domain of physicists but ordinary household knowledge. But we can’t do any of these things, and so we have science, to deduce about the gargantuan what we, with our infinitesimal faculties, will never sense. If people fear big numbers, is it any wonder that they fear science as well and turn for solace to the comforting smallness of mysticism?