Spoken language may be too vague, while mathematics can be too exact, either of these extremes can be less productive.
Math is a great tool for crunching relations/abstractions, similar to how computers are good at crunching numbers. Though, it is too exact to let room for any serendipity. Serendipity helps a lot when exploring.
I don't see mathematics as a seeking tool. Only using it occasionally, as a confirmation tool to check if an idea is correct, or as a dismissing tool, to get rid of those too seductive hypotheses that stick to mind.
Then, I don't know enough mathematics. It's hard to learn, hard to handle, mistakes are easy to make but hard to spot, it is crucial to properly encode, then decode/interpret the results, etc.
Most of all, it's an axiomatic system, that started from the real world. Remember it all started from counting the sheep and measuring the land. That also means it will not work outside its axioms. There are always tacit assumptions, even in math axioms. Having in mind the way geometry started from measuring flat agriculture land, and that led to Euclidean geometry. If axioms are even slightly modified, we get new theorems and new mathematics, like the non Euclidean geometry. Not a big deal now, looking in perspective, but a big deal if back then one would try to use Euclidean geometry to prove/disprove some non Euclidean hypothesis. One would have get a big no, because of using the wrong set of axioms.
I guess these are all epistemology 101 and studied systematic there, though I never managed to follow any book or lectures about epistemology, looked very outdated. I see human mind as a fuzzy lookup table, where the lookup table entries were filled up in the very early years, with notions we learn by interacting with the world. Those are the basic concepts in which we understand everything else. Later we learn to associate the most common entries in our lookup tables with words. Filling the table happens in the very first years of life, before self-awareness emerges. The table will keep updating for the rest of the life, but at a much slower peace. When a fact matches very well the lookup table entries, we are in the ordinary, expected, or maybe boring situations. When we encounter something for which we don't have a matching entry yet, we tend to reject that, or to take it as a "mind-blowing" new something.
All we do, and feel, and think is in terms of our inner lookup table,
plus some internal noise. The inner lookup tables being fuzzy and noisy, can sometimes lead to unexpected behavior or to creativity.
Sounds a lot like tokens from ChatGPT, but the fuzzy lookup tables idea came to me from hardware, not software. I've start playing with the lookup table analogy a couple of decades ago, probably inspired from the FPGA architecture, where everything is a lookup table.
Even funnier when it comes to awareness. I think what we call self-awareness is the ability of computing assisted-simulations of the world in which we live in, based on two things:
- the already existing entries in the fuzzy lookup tables
- the live sensory input data stream we have from our senses
By assisted-simulation, I mean the projections of the outcome is based on the lookup table, and the sensory input is used as a feedback. Sensory input (e.g. vision, hearing, tactile, etc) is what we compare against, in order to check how accurate the prediction/simulation was. Then adjust the simulation accordingly. If we are left out without any sensory input, we start hallucinating (e.g. when sleeping, or under certain drugs, or in isolation tanks).
In fact, the more advanced an animal is in the tree of life, the more aware and the more capable is at computing the surroundings. Reaction times are very slow, so animals got themselves good at predicting the future. I believe our day to day life is a living simulation, a dream that is continuously adjusted, adjusted such that the simulation won't diverge too far from the sensory input stream.
We navigate reality, and react to what our mind
predicted, rather than to what our senses read back from the environment.
Sorry for going with the flow with the offtopic, it's hard to be concise and convincing at the same time. All these might be as well nothing but too much noise in the lookup tables.
Not trying to dismiss math, mathematics is a must have for rigorous work. Though, when seeking/exploring at leisure, I found handling concepts from that fuzzy lookup table more fun to work with than math. It's way faster to process inner thoughts, and more productive at generating new ideas. Then triage later which ones may worth investigating further.
Many such generated ideas will be way off, and the internal lookup table keeps updating according to what it is exposed-to the most (that's a pitfall of our inner working, learning is always on, so a lie repeated a thousand times becomes truth). So it's a risk that, eventually, some of the false ideas generated by words instead of math will be assimilated as truth in the own lookup table. And just like that, one might start wearing pointy hats and pretend to be Napoleon.
That's why one should never skip the mathematical proof.